Tauberian Theorems
Norbert Wiener
The Annals of Mathematics, 2nd Ser., Vol. 33, No. 1. (Jan., 1932), pp. 1-100.
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Sun Sep 2 02:38:52 2007
TAUBERIAN THEOREMS..
BY NORBERTWIENER.
-
Page
INTRODUCTIO Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER
I. THE CLOSUREOR THE SET OF TRANSLATIONS
OF A GIVEN HUNCTION 7
1. Closure in class La . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Closure in class L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. A sub-class of Ll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER I1. ASYMPTOTICPROPERTIESOR AVERAGES. . . . . . . . . . . . . . . . . . . . . . . . 4 . Averages of bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Averages of bounded Stieltjes distributions . . . . . . . . . . . . . . . 6. Averages of unilaterally bounded distributions and functions
OF SERIES AKD
CHAPTER I11. TAUBERIANTHEOREMSAND THE CONVERGENCE
INTEURALS
............................................... 7. The Hardy-Littlewood condition . . . . . . . . . . . . . . . . . . . . . . . . . 8. The Schmidt condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER IV . TAUBERIANTHEOREMSAKD PRIME NUMBERTHEORY. . . . . . . . . . . 9. Tauberian theorems and Lambert series . . . . . . . . . . . . . . . . . . 10. Ikehara's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 CHAPTER V . SPECIALAPPLICATIONSOF TAUBERIANTHEOREMS
. . . . . . . . . . . . . . . 60 11. On the proof of special Tauberian theorems . . . . . . . . . . . . . . 50 12. Examples of kernels for which Tauberian theorems hold ... 52
13. A theorem of Ramanujan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 14. The summation of trigonometrical developments . . . . . . . . . . 55 15. Young's criterion for the convergence of a Fourier series . . 57
16. Tauberian theorems and asymptotic series . . . . . . . . . . . . . . . . 60 CHAPTER VI . KERKELSALMOSTOF THE CLOSEDCYCLE. . . . . . . . . . . . . . . . . . . . . . 62 17. The reduction of kernels almost of the closed cycle to kernels
of the closed cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18. A Tauberian theorem of Hardy and Littlewood . . . . . . . . . . . 19. The Tauberian theorem of Bore1 summation . . . . . . . . . . . . . . CHAPTER VII . A QUASI-TAUBERIAN
THEOREM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. The quasi-Tauberian theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. Applications of the quasi-Tauberian theorem . . . . . . . . . . . . . . CHAPTER VIII. TAUBERIANTHEOREMSAND SPECTRA. . . . . . . . . . . . . . . . . . . . . . . . . . 22. A further type of asymptotic behavior . . . . . . . . . . . . . . . . . . . 23. Generalized types of summability . . . . . . . . . . . . . . . . . . . . . . . . 90 24. Some unsolved problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 INTRODUCTION .
Numerous important branches of mathematics and physics concern themselves with the asymptotic behavior of functions for very large or very
* Received July 20. 1931.
1
1
2
N. WIENER.
small values of their arguments, or of certain parameters. Statistical
mechanics is that branch of mechanics which concerns itself, not with an
individual dynamical system of a finite number of degrees of freedom, but
with the asymptotic behavior of dynamical systems as the number of
degrees of freedom increases without limit. The analytical theory of
numbers is likewise concerned with the behavior of assemblages of whole
numbers as the size of these assemblages increases. To this domain of
ideas belong asymptotic series in analysis, and a whole order of concepts
clustering about the operational calculus of Heaviside and the Fourier
series and integrals.
The Fourier integral is of peculiar interest in the study of asymptotic
problems. If f (x) is a function of Lebesgue class L2, by a theorem of
Plancherel,' there is related to it another function g ( u ) , also of L,, defined
by the equation
(0.01)
g!u) = 1. i. m. ---- J A
A+a
1/%
-Af(
x) eiux cix
(where 1. i. m, stands for "limit in the mean"), such that
and
(0.03)
1
PA
f ( x ) = 1. i. m. --a+m 1/gJ - A g (
U ) e-iux
d7h.
The functions f ( x ) and g(zr) have the reciprocal relation, that asymptotic
properties of each correspond to local properties of the other. They are
known as Fourier transforms of one another. It is easy to see that
f (2-4-3,) and g(u) eWiuA are likewise Fourier transforms of one another.
There is a large class of asymptotic problems in which the asymptotic
property to be investigated is connected in some obvious and simple way
with the entire class of functions f (x+Iw). Since the Fourier transforms
of the functions of this class only differ by factors eciuA, independent of
the particular function f ( x ) , Fourier transformation is here a peculiarly
useful tool. An example of such a problem is the investigation of the
asymptotic behavior of the integral
Many problems which on first investigation do not appear to be concerned with integrals of the above sort may be put into such a form by
an elementary transformati011 of variables. F o r example, if we put
' Cf. Plancherel (l),Titchmarsh (3),
liellin (l),Wiener (7).
Cf. Bibliography on p. 94.
-
TAUBERIAN THEOREMS.
the integral in question assumes the form
Thus the study of the asymptotic properties of this integral also fall under
those accessible through Fourier developments.
I n 1925 a paper by Robert Schmidte appeared in which the class of
theorems known as Tauberian was brought into relation with the asymptotic
properties of integrals of this type. Tauberian theorems gain their name
from a theorem published by A. TauberS in 1897, to the effect that if
and
(0.08)
then
(0.09)
This is a conditioned converse of Abel's theorem, which stated that (0.07)
follows from (0.09), without the mediation of any hypothesis such as (0.08).
Such conditioned inverses of Abel's theorem, and of other analogous theorems which assert that the convergence of a series implies its summability
by a certain method to the same sum, have been especially studied by
(3. H. Hardy and J. E. Littlewood, and have been termed by them Taz~be~ian.
It is the service of Hardy and Littlewood4 to have replaced hypothesis
(0.08) by hypotheses of the form
or even of the form
(0.11)
nun>-K.
The importance of these generalizations is scarcely to be exaggerated.
They far exceed in significance Tauber's original theorem. The work of
Hardy and Littlewood, unlike that of Tauber, makes very appreciable
demands on analytical technique, and is capable, among other. things, of
supplying the gaps in Poisson's imperfect discussion of the convergence
' Schmidt (2). Tauber (1). 'Hardy (2), Littlewood (I), Hardy and Littlewood (4)' (10). (13), (18) ete. '1
4
N. WIENER.
of the Fourier series. F o r these reasons, I feel that it would be far more
appropriate to term these theorems Hardy-Littlewood theorems, were i t
not that usage has sanctioned the other appelation.
As we said, Tauberian theorems reduce to theorems on the conditioned
equivalence of the asymptotic values of certain averages. The auxiliary
conditions of Hardy and Littlewood, together with certain more extended
conditions given by Schmidt5 and discussed by Vijayaraghavan6 and S z & s ~ , ~
become restrictions on the magnitude of the function or mass distribution
of which the average is to be taken. An even further reduction eliminates
this function from all consideration, and finds the essence of the Tauberian
theorem in the liuear properties of the weighting functions used in determining the average in question. The linear properties which are of
importance concern the closure of the set of functions derivable from a
given weighting function by translation, and the first chapter of this
monograph is devoted to the study of such closure properties. The second
chapter is occupied with the formulation and proof of the fundamental
Tauberian theorems concerning averages, and the third with the transformation of these theorems into a recognized Tauberian form.
Some of the most interesting applications of Tauberian theorems have
been in the analytic theory of numbers. Here two different avenues of
approach are possible. Hardy and Littlewoods have shown that the prime
number theorem, to the effect that the number of primes less than n is
asymptotically nllog n , is entirely equivalent to the Tauberian theorem
concerning Lambert series, that if
(0.1 2)
and
(0.11)
then
This theorem falls under the category of theorems demonstrable by the
methods of the present paper, and involves no further information concerning
the Riemann zeta function t; (of zi) than that this must be free of zeros
on the line o = 1. This theorem was already contained in the author's
first note on his method, dated 1928, and establishes the usefulness of
Lambert series in the proof of the prime number theorem, which had
Schmidt (1) and (2).
Vijayaraghavan (1) and (2).
S Z ~ S (Z1) and (2).
Hardy and Littlewood (11).
TAUBERIAN THEOREMS.
5
frequently been questionedg. The proof was however needlessly complicated
by the fact that after a good deal of labor was spent in transforming the
author's average theorem into recognizable Tauberian form, this form did
not appear to be especially direct in its relation to the prime number
theorem. The present memoir gives a much more direct proof, starting
from a theorem of the average type.
The other approach to the prime number theorem eniploys a theorem
of S. Ikeharalo, which is itself a generalization of a theorem of Landau.
Landau" showed that if 2 a, n-z is a Dirichlet series with positive
coefficients, representing a function q(s) analytic on and to the right of
W (z) 1, except for a pole of order 1 at x = 1, for which the residue
is A, and if the function in question is O(lx for W (x)2 1, then
-
(0.13)
Ik)
lim
n+m
-n1 2o% al = A.
Ikeliara proved by a Tauberian theorem that the condition that q(x)= O(lzlk)
is inessential. By applying the theorem to q (x) = - 5' (x)/C(x)
it is at
once seen that the prime number theorem again follows from the fact that
the Riemann zeta function has no zeros on the line W(z) = 1.
Chapter V is devoted to miscellaneous applications of Tauberian theorems.
Among these perhaps the most important are to trigonometric developments
and their summability. Here a theorem of Ramanujan" on Fourier transforms is of service. A particular application of Tauberian theorems to
the criterion of Young1' for the convergence of a Fourier series is due
to Dr. Littauer14, and is here discussed. TVe also take up the application
of Tauberian theorems to asymptotic series and Wintner7s15work.
There is a further field of application for Tauberian theorems where wt
need to introduce something like Robert Schmidt7s16notion of "gestrahlte
Matrizen". This we discuss in Chapter VI. In this c,ategory we find
certain theorems of Hardy and Littlewood17 related to Abel summation, as
also the Tauberian theorems that concern themselves with Bore1 summationls.
--
Wiener (4).
Ikehara (1).
" Landau (3).
l2 Hardy and Littlewood (8).
l 3 Young (1).
l4 Littauer (2).
l5 Wintner (1).
l6 Schmidt (2).
l 7 Cf. Hardy and Littlewood (lo), (13), (18); Karamata (I), (2), (3), (4) ; Szhsz (I), (2) ;
Doetsch (3); etc.
ls Cf. Hardy and Littlewood (2), (3) ; Doetsch (4); Schmidt (1) ; Vijayaraghavan (1) ;
Wiener (4).
lo
6
N. WIENER.
In Chapter VII, me discuss certain theorems not strictly of a Tauberian
nature, in that they involve no positiveness or boundedness condition. We
here follow an earlier paper1' in which the author proved the HardyLittlewood necessary and sufficient condition for the suinmability of a Fourier
series. There our result was not a "best possible" one, as the work
of BosanquetZoand Paley has demonstrated. At
our method has
succeeded in yielding their strict theorem.
Chapter VIII is devoted to the development of certain Tauberian theorems
intimately connected with our generalized harmonic analysis8'. These suggest
certain new definitions of summability, and we discuss definitions of this sort.
The genesis of the present paper may deserve a word of comment. In
the preparation of a previous investigation on generalized harmonic analysis,
the author found himself obliged to make use of certain theorems communicated
to him by Mr. A. E. Ingham and of a Tauberian nature. A correspondence
arose with Professor Hardy, finally resulting in a number of papers of
a Tauberian character. While interested in theorems of this type, the
author had the work of Dr. Robert Schmidt brought to his notice, and at
Dr. Schmidt's suggestion, began to search for a method of combining the
trigonometric attack of his own earlier papers with the generality of the
Schmidt standpoint. I t was only by a radical change in the manner in
which trigonometric methods were applied that this attempt succeeded.
This change consisted in a logarithmic change of base before the introduction of the harmonic analysis. It was quite along lines contemplated
by Schmidt, who correlated with his Tauberian theorems a certain moment
problem. Schmidt, however, found in the problem of uniqueness the significant aspect of this moment problem. The present author looked for it
rather in the problem of the existence of a solution together with certain
associated problems of approximation. I t was Dr. Schmidt himself who
furnished the experimentum crucis which established the greater scope of
the methods of this paper. He suggested that the Tauberian theorem for
Lambert series had resisted his methods of proof, and that it might be
desirable to try on it the edge of any new method.
The origination of a new method is but the prelude to a large amount
of detailed application before its power can be judged or its limitations
defined. In this task the author has had the valuable aid of his students,
Drs. S. B. Littauer and S. Ikehara. He also wishes to express his gratitude
for the criticisms of Professors Hardy and Tamarkin a t more than one
stage of his work.
Cf. Hardy and Littlewood (12); Wiener (5). Bosanquet (1); Paley (1).
*'Wiener (7). l9
20
TAUBERIAN THEOREMS.
CHAPTER I. THE CLOSURE OF THE SET OF TRANSLATIONS OF A GIVEN FUNCTION. 1. Closure in class L,. Let f (xj be a function, real or complex,
defined for all real arguments over (- oo, m). Let it belong to Lebesgue
class L,-that is, let it be measurable, and let
be finite. We shall term the class of all functions f ( x + I ) for all real
values of I the class of translations o f f ( x ) . We wish to know when
this class is closed or complete-that is, when it is possible, whenever
a function F ( x ) from L2 is given, and any positive quantity s, to find
a function Fl ( x ) of the form
n
(1.02)
F,(x)
=
CA~~(X+I~),
1 such that
(1.03)
:l
[ F ( x )-F1(x))l2dx
s.
The situation is governed by the following theorem:
THEOREM
I. The necessary and sufjcient condition for the set of all
translations o f f ( x ) to be closed is that the real zeros of its Fourier transform
should form a set of zero measure.
The existence of this Fourier transform results from a familiar theorem
of Plancherel.' This theorem further requires that
and that
(0.03)
I
JA
f [x) = 1. i. m. -----
VG
A+,
+
-*g(
~ ) e - ~ d a .
.
The Fourier transform of f !x I ) is eciuAg (u)
On account of (0.02); properties of convergence in the mean are conserved under a Fourier transformation, for these concern merely the integral
of the square of the modulus of the difference of two functions, which is
equal to the integral of the square of the difference of their Fourier
transforms. Accordingly, Theorem I is a corollary of:
8
N. WIENER.
LEMMA
I a . I f g(u) is a function belonging to L2, then tlze set of
fulzctions e - i 2 1 R g ( ~ )is closed tohen and only when tlze mSos of g(u) form
a set of zero nzeasul-e.
Since every zero of g(u) is also a zero of e-iuR g ( u ) , the necessity of
this condition is a t once obvious, for if
is any linear combination of the functions e--iu""gu), then
must equal or exceed the measure of the set of the points between A and
B where g ( ~ t v) anishes. That is, if the measure of this set is not zero?
the function which is 1 between A and B and zero elsewhere cannot be
a limit in the mean of functions Gl(u).
Now as to the sufficiency of the condition in Lemma I a : let
(1.06)
Then
This is an absolutely convergent integral, and hence a limit of a sequence
of polynomials
(1.08) e-iuRn
CAf1,
converging boundedly, and uniformly over every finite interval. Hence
-2
e
.unn
A Y(u)~(u)
is a limit in the mean of such functions as those in (1.04).
Now let q ( u ) be a quadratically summable function satisfying the conditions
-iunn
(1.10) Since
S _ B * ~ ( u ) t ( u ) g ( u )Ae d u = 0 .
[ n = . . ., -2, -1,
q (u)gp (u)g(u) is absolutely s ~ r n m a b l e we
, ~ ~shall have
(1.11) +(u)Y(u)g(u)= 0
22This follows from the Parseval theorem for Fourier integrals.
0 , 1 , 2,
...I
TAUBERIAN THEOREMS.
9
except over a set of measure zero.83 If g(u) does not vanish except over
a set of measure zero, since y (u) has no zeros, q ( u ) vanishes over
(-A, A) except over a set of measure zero. Thus if g(u) # 0 , there is
no quadratically summable function not almost everywhere zero, ortho-iunn
gonal to every function e A y (u)g(u) over (-A, A). This means that
every function of La vanishing everywhere outside ( - A , A) is a limit
-imn
in the mean of polynomials e4 in e A y (u)g(u) and hence in e-i'b" g(u) .
Every function of Lp is however the limit in the mean of a sequence of
functions of L, vanishing for large arguments. Thus Lemma I a and
Theorem I are established.
2. Closure in class L1. The Lebesgue class L1 consists of all measurable functions f (x) which are absolutely integrable over (- co , a).
If f (x) is such a function, its Fourier transform
of course exists, and is bounded and continuous. As before, the Fourier
transform of f (x A) is e-iuAg(u).
We shall say that a class C of functions y ~ ( x )of LI is closed Ll if,
whenever F(x) is a function of L, and E is a positive number, there is a
polynomial
+
?I
(2.02)
Fl (x) =
2 Ak Y Ak (x)
1 of functions of C, such that
We shall prove the following theorem:
11. If f ( x ) i s a fzinction of L1, a necessary and szcf$cient
THEOREM
condition for the set o f all translations o f f (x) to be closed L1 i s that its
Fourier transform
1 (2.01)
g (u) =
f (x)eiux d x
should have n o real zeros.
XL
---- -.--- - --
23This follows from the Riemann theorem on
Fourier series.
24Here we make use of the familiar theorem
L, is given, every function of L, is the sum of
functions of the class and a function orthogonal
the unicity of the function with a given
that if any linear class of functions of
a function expressible in the mean by the
to all functions of the class.
10
N. W I E N E R .
The necessity of this condition is again obvious. If g(zcl)
same is true for the argument zt, of the Fourier transform of
=
0 , the
Let f l (x) be a function of L, with a Fourier transform g1 ( x ) for which
=
0-I
sl(z4)
Then
This proves that f1 is not a limit in the mean (L1)of ariy sequence of Flls.
To prove the sufficiency of the colldition of Theorem 11, we shall have
recourse to a sequence of lemmas.
LEMMA
IIa. I f
and
m
zihlai I ? < = ,
CC
2l~=
l~
A~
< w/,
-m
(2.06)
=
-m
and i f '
tlzen
The proof is immediate, for
2I
(2.09)
-m
c.
=
2 2 nu L-I,
I%=--m
m
=z
ja,,j
-50
k=-m
i
5
I
2 2 1 ax 1
,I=-m
lc=-m
I h,-x I
00
2
I b,,/ = A B .
-m
LEMMA
IIb. I f f ( x ) is a ficnction of period 2 z, and i f at every point y ,
there is a f ~ t n c t i o n &(~x ) , coincident with f ( x ) ovev some intwcal ( y - ~ ~y ,+ ~ ~ ) ,
of period 2 z , and srhch that its Fozirier series converges absolutely, then the
Fourier series o f f ( x ) cofzcerges absokctely".
25 TTe shall say that a Fourier series converges absolutely when the sum of the moduli
of the coefficients converges, the series being represented in complex exponential form.
This is of course equivalent to the absolute convergence of the series in this form for
any single real argument.
TAUBERIAN THEOREMS.
11
To prove this, let us reflect that by the Heine-Bore1 theorem, any period
of f(x) may be covered by a finite number of overlapping intervals
(y-F,,
y4-eu). Let such an interval be (A,B), let its neighbor to the
left have its right-hand end-point a t C, and let its neighbor to the right
have its left-hand end-point a t D. There mill be no loss in generality if
D < B . Let us define y,(x) by
we suppose that d;C<
I-.*.
c-d
U-D
,
[
'
1 D S'~:<B]
-J-Sx<Cl
It is easy to show that the Fourier series of y,(x) converges absolutely.
Thus the Fourier series of
(2.11)
Y?, ( 4 f(5)= Yy (4 J; (2)
converges absolutely, by Lemma IIa. However, if we add together the
functions y, (x) for a set of intervals completely covering a period of f(x),
the sum will be f (c)itself. Thus f(x) will be the sun1 of a finite number
of functions with absolutely convergent Fourier series and must itself have
an absolutely convergent Fourier series.
then tlze Folcvier series o f 11f (x) co?tceyqes absolzctely.
W e may write 1 1
-1
tc
f ( 4 - no
n, einx
a,,eiptx
+2
-00
+2
1
That this is more than formally true, and that l/j'(x)is continuous,
results from the fact that the geometrical progression
12
N. WIENER.
converges. Term by term, series (2.15) is greater than or equal to the
sum of the sums of the moduli of the coefficients of the Fourier series of
the successive terms of (2.14), and hence is greater than or equal to the
sum of the moduli of the coefficients of the Fourier series of 1 lf (x).
L E M M A I I ~Let
.
(2.16)
and let
(2.17)
Let
+
m e n there is a neighborlzood ( y -E , y s) of y and a function f y ( x ) with
absolutely co?zvergent Fourier series sztch tlzat over ( y -E, y s) ,
+
and that the Fot~rierseries of f,(x), let u s say
m
2
(2.19)
has tlze property that
C,l
-m
ei~~z
There is manifestly no restriction in taking y to be 0 . Let us introduce the auxiliary function
(2.201)
This will have the Fourier series
3&
m einx + e-inx
--
2m
+ 2 rnD-E
(cos e n - cos 2 s n ) .
1
If we represent the function { f (x) y , (z)+J ( 0 ) [ l - ~ ,( x ) ] }by the Founer
series
TAUBERIAN THEOREMS.
we shall have
-t
ak-m
(2.203)
=
n,eE
>
m
cosak-
ao+2(n-i;-t-ak)[1+
cos2rlc
nksE
3~
2rr
].
Hence
and
-M
Here
2" means
that the values 0 and nz are omitted.
TVe have
1
c o s ~ n - cos2~n)
E 92
2 sin --- sin --2
2
76,'E
3t
2 .
nn2, '
and
(2.21)
=
/J
a-m.
d2
j
COS EX - COS 2EX
zx2E
This formula may be applied in the limiting form whenever m = n or
m = 0. It results from (2.205) and (2.206) that for all sufficiently small E
14
N. WIENER.
and from (2.205) and (2.21) that for all sufficiently small
6
Combining (2.205), (2.22) and (2.23), we get
which with (2.204) yields us
If we take
(2.26)
.f;/(x) = f (x) YE(X)+f (0) (1- Y E (XI),
Lemma I I d is established.
As a direct consequence of Lemmas IIb, I I c , and I I d we obtain:
LEMMA
IIe. If f (x) i s a functiolz with a n absolzitely co7zz~ergentFozcrier
series, whicll n o w h e ~ evanishes for yea1 argume7zts, llf (x) has a n absolutely
convergent Fourier series.
To prove this let us note that, in every neighborhood, by IId, f (x)
coincides with a fnnction which by I I c has a reciprocal with an absolutely convergent Fourier series. By l I a , since in every neighborhood
l / j ( x ) coincides with a function with an absolutely convergent Fourier
series, its Fourier series converges absolutely.
IIf. L e t f (x) be a function with a n absolutely co?zvevqent Fourier
LEMMA
series over tlte interval (- z, z ) , and vanishing over na~qhborlzoodsincluding
rr and -n . Let f (x) v n n i s l ~ everywhere outside (- x, z ) . Let
T h e n the integral
zvill conver,qe. Conversely, i f f (x) vanishes ocer tlze region indicated, and
(2.28) converges, its Fourier series will converge absoltctely.
Let E be such that j (x) has no non-zero, values outside ( - z + 2 ~ , z-26).
Let us define y (x) by :
TAUBERIAN THEOREMS.
Let
(2.17)
We shall have
and hence
where the rearrangement is possible since both integral and sun1 form part
of an rtbsoluteljr convergent double summation process. Then
will satisfy the condition that (2.28) shall be finite, and ecluation (2.27).
To prore the converse part of Lemma IIf, let us notice that as a result
of the convergence of (2.28),
converges.
Now,
Thus the Fourier series of
2 sin xI2
f(x>
converges absolutely.
(2.36)
X
Since t'he same is true of the Fourier series of
X
2 sin 212 Y (4
16
N. W I E N E R .
as may be determined by a simple computation, i t follows from Lemma I I a
that the same is true for the Fourier series of f ( x ) .
LEMMA
I Ig. Let f ( x ) be a filnction of class L1. T l ~ e n
This is a well-known theorem and is proved in Hobson's Theory of Ft.lnctions
of n Real Vnriable.
LEMMA
I Ih. Let f ( z ) Be a function of class L, . Then i f 0 ( y ) is any
ficnction of class L,, difa-ing front 0 only over ( - E , E ) ,
The proof of this lemma merely depeiids on the inversion of the order
of integration in an absolutely convergent double integral.
IIi. Let f ( x ) be a fzbnction of class L1. Then
LEMMA
100
1
m
(2.39)
lim
9z+m
1
3m
f ( x )- --
f(x
z 9%
+y) sin2y2 y
92
d y d x = 0.
This is a theorem of the Fej6r type. It depends on the splitting of
Qo'
into the sum
(y) =
sin2n y
7-
a,(y) + Op( y ) where
Lemmas I I g and I I h enable us to show that
00
(2.41)
linl
n+w
while
1
(2.42)
1
1
S_OOf(~)-nlrS_mf(x+l/)
+
y
I
~ I ( Y ) ~a xY = 0
y
713m
1
00
TAUBERIAN THEOREMS
Now
=
.
0 (n-ll2 log n)
Hence
00
(2.422)
and Lemma I I i is established.
We are now in a position to proceed to the proof of Theorem 11. Lct
F ( x ) be any function of L,. By Lemma IIi, we can find a function
F 7 ( x ) of the form
such that
:
l1
(2.44)
F(x)-F7(x) 1 dx<q
The Fourier transform H , ( u ) of this function FT,(n.)is
00
(2.45)
00
sin' 'n y
@MY
dl
=
and will vanish for all values of its argument 2.c larger than 2 n in modulus.
The same will be the case wit,h the Fourier transform HO(u)of
18
N. WIENER.
which mill vanish for values of its argument greater in modulus than 492,
and only for such values. If we expand these Fourier transforms in
Fourier series over (-8 1 2 , 8 n ) , by Lemma IIf, these series converge
absolutely.
By an argninent substaritially identical with that used in proving Lemma I I b ,
we see that in the neighborhood of every point of (- 392, 3 n ) , there is
n function coinciding locally with I / H, (u), and with an absolutely convergent
Fourier series. It follows from Lemnla 1 I a that the same is true of
HI (zi)IH2(zt) over the same interval. Since 0 enjoys this property also,
the function which is HI (21) I H, (u) over (- 3 n , 3 n) and 0 over the rest,
of a period (-8n, 8 n ) has an absolutely convergent Fourier series, and
hence, by Lemma IIf, an absolutely convergent Fourier integral. That
is, we may write
(2.47)
HI (zc) = Hz (u)H3 (26)
where H3(tt) is of the form
and
(2.49)
converges. W e shall then have
Here we may replace eitLx by any function with an absolutely convergent
Fourier integral, and hence by functions positive over an arbitrarily small
region and zero everjrwhere else, This can only be if
19
TAUBERIAN THEOREMS.
Here
Rj7
ty ( o ) is
absolntely integrable.
Lemma I I h and Lemma IIg,
Combining this with (2.52) and Lemina IIi, me obtain Theorem 11.
With only a slight modification of detail, we have proved:
THEOREM
111. Let f i ( x ) and fi(x) be tzuo fitnctions of I;, Let
.
and
(2.55)
Let tlze set o f points w h e ~ egi(zt) 0 consist only o f inner points o f the set
wllere g, ( 1 0 0 . T l ~ mi f s >0 tlzel-e i s n polynomial
+
sllch that
(2.57)
On the basis of Theorem 111, we are in a position to prove the general:
IV. Let P be a class o f ftinctions o f L1. T h e n a necessaq
THEOREM
( ~ n dSZLf$cie?zt condition for the class 2, , contai?zing all functions f ( x Lj
zuheneve~f ( x ) belongs to 2 , to be closed L 1 , is thctt there slzoz(ld be n o renZ
sero common to all tlze Foztrier transfornzs o f ficlzctions o f 2.
The proof of the necessity of the condition of Theorem IV does not
differ from the similar proof in the case of Theorem 11. As to the sufficiency, it follows a t once from Theorem I11 and the fact that the Fourier
transform of a function of L, differs from 0 on an open set, that for
every 2 1 , an E may be assigned, such that
+
20
N. WIENER.
is a function approxiinable L1 by polynomials in functions of 2,. Let
(-- U, U ) be any interval of frequencies. By the Heine-Bore1 theorem,
this may be overlaid with a finite number of overlapping intervals U9,- En,
D n t ~ n . Let US form
dqs sin2E . x
(2.59)
x
2
for this set of intervals. This will be a function of L,, approximable L,
by polynomials in the function of 2,, whose Fourier transforni may be
shown to be
(2.60)
~ " (41 - 2 1
+F
n
(
)[sgn(l-eI
1
"+'
en
I
and to have no zeros over (- U, U). Thus by Theorem 111, every function
of L, with a Fourier transform vanishing outside (- U + n , U-n) will
be approximable L1 by polynomials in the functions of 2,. At this stage,
the introduction of Lemma I I i serves to complete the proof of Theorem IV.
I n Theorems I1 and IV, the necessary part is nearly trivial and all the
difficulty resides in the proof of sufficiency. There are certain applications, however, where it is precisely the necessary part of the theorem
that is significant. W e may prove a function to have no zeros if it is
the Fourier transform of a function of L1 the set of whose translations is
closed L,. To establish this closure is indeed more than is needed: it is
enough to produce for each u among the class of functions to which we
may approximate L1 by polynomials in the translations of our given functions, a t least one function whose Fourier transform does not vanish for
the argument 2 1 .
It may be shown if F ( z ) is a non-negative function for which
(2.61)
and
(2.62)
then
(2.63)
This results from the fact that for
-B
5L 2 B ,
Accordingly, if for every value of B and E , we may find a function of
this sort which may be approximated L1 by polynomials in the translations of f (x), the Fourier transform of f (x) has no zeros. As in the
21
TAUBERI AN THEOREMS.
case of Theorem IV, the function f ( x ) may be replaced by the class 2 ,
and the class of translations of f ( x ) by 8 , . Then the condition that the
Fomier transform of f ( x ) shall have no zeros is to be replaced by the
condition that there shall be no zero common to all the Fourier transforms of functions of 2 .
This method is a conceivable one for the investigation of the zeros of
functions with known Fourier transforms, and might be applied to the
study of the zeros of the Riemann zeta function. So far it has yielded
no results. Under certain conditions, the Taylor series may be introduced
to reduce the question of the closure of the set of translations of f ( x )
to that of the closure of the set of functions
(2.65)
f
'")
.
(x)
This matter has not yet, however, been subjected to an adequate investigation.
3. A sub-class of L,. I n connection with Stieltjes distributions of
niass, we shall have to consider ('kernels" of class L1, fulfilling a certain
more restrictive condition. This condition on a measurable function f ( x )
is that the series
-
k--m
lim
kA+BCx:(kf l)A+B
1f
(4i
shall converge. W e may easily verify that this condition is in fact independent of A and B, and that it can only be fulfilled by functions of L,.
With it goes a certain definition of the "distance" between two functioils
f ( x ) and g ( x ) , this "distance" being defined as
This notion of "distance" has a very important difference from
and
in that i t is not true that if (3.01) is finite, then
(3.05)
lim
e+O
2 k slim
If(x+~)-f(x)i=O.
xjktl
k=-m
22
N. WIENER.
Accordingly, if we are to prove theorems analogous to those in the last
paragraph without a radical change in method of proof, we must introduce a restriction which will make (3.05) hold. Otherwise we shall be
unable. to establish the analogue of Lemma IIg.
A condition of the desired sort is that f ( x ) shall be continuous. It will
then be uniformly continuous over any finite interval, and me shall be
able, first to choose A so large that
(3.06)
max
-A-k-1zxz-A-k and then to choose 7 so small that for - A -
It will follow that for 0 4 ql
15x
2 A + 1, 0 < g, < g,
< 11,
We shall call the class of all continuous functions for which (3.01) is
finite, the class M I , and we shall say that a class 2 of functions of MI
is closed MI, if whenever F(x) belongs to MI and s > 0, there is ii polynomial
1L
27 Ahfh(x)
(3.09)
Fl ( x ) = h = l
in functions of the class 2 such that
max F ( x ) - F l ( x ) i & .
k=-m
kzx2kfl The following theorem is valid:
THEOREM
V. L e t f ( x ) belong to L1. Tlten a necessary and szcf$cient
condition that whewever fi ( x ) i s a function o f MI and E > 0 , there slzall
exist a ficnction ~ ( z o) f MI such that
i s thnt for n o real
11
I
/-6
m
The p r o d differs in no essential respect from that of Theorem 11. Let
us note that if f,(x) belo~lgsto L1 and
23
TAUBERIAN THEOREMS.
then & ( x ) belongs to HI.To see this, let us put
1
(3.14)
& (x)e-iux d x
=
gz ( Z L ) ;
( t i ) is bounded and differs from 0 only over a bounded range, and hence
belongs to L,. Thus
g,
exists as a function of L2. Further
y2( e l )
=
1
1. i. in. -=
B-tw 1 / 2 n
'B
( x )e-iux d x
J9
exists arid belongs to L 2 . If we put
where # belongs both to L2 and Ll, it follows readily from (3.14) and
(3.16) that
We can choose a set of F ' s vanishing outside a given finite interval and
closed L, and hence L, over this interval. Then over this interval, which
is arbitrary, J2 and f3 can differ a t most on a null set. Thus f 2 belongs
to L,.
Now
(3.19)
92 ( 2 4 ) = g2 (4
W(z4)
where
(3.20)
Thus by the Parseval theorem,
24
N. WIENER.
From this it follows that
2
(3.22) S=-m
W
1
max l ~ ( . z ~ ~ ~
h:xzk
-m
-1
/=m
and that Ji(z), which obviously can be modified so as to be continuons,
belongs to MI.
I t only remains to prove the analogue of Lemma I l i , and to show that
every fullction of class ,$Ilmay be approximated with any degree of
accuracy in the Jll sense by functions with Fourier transforms vanishing
for large arguments. This follows exactly tlie lines suggested by IIi.
We wish to show, that is, that if f (x) belongs to J f i ,
AS before, we put
(3.222)
1 - V 1 ;
@,(y) =
The proof that
(3.224)
lim (f(x),
S + 0 0 ,
1
Lrn
[ j y ~h7-l12]
m
f(z+r)
@I
(Y) a x )
0
;
i
follows exactly the lines of that of (2.41), for the analogue of Lemma I I g
has already been shown to be true, while that of Lemma I I h is proved
by the same arguments of absolute convergence as the lemma itself.
Again,
W
2 2 k=-m
2
nlax
kzxzkf I
1 f (x)
1
lim
1
00
-
and this, by (2.421), is zero. This completes the proof of (3.221).
An extension of Theorem V in the direction of Theorem I V is tlie
following:
25
TAUBERIAN THEOREMS.
THEOREM
VI. L e t 2 be n class o f f i ~ n c t i o n s o f L1. L e t El be tlze class
of all fiuxctions o f tlze fornx
$
Ah( y ) a,,(1:-1y) d y
JK
W
to1lel.e j; , . . Jlv belon.9 to 2 a n d a l , . ., a N to ilil. TIze11 HIi s closecl
llfl zclten a n d o n l y zolzen there i s t2o real valzle o f zc ~olzichi s n ze9.o co??znzon
to all tlze fzinctiorzs
a ,
sW
L--m
(3.24)
1/27
(J)
(:iuZ
( 1 ~
c.o~.~.esponcli?~,
to firnctio~lsfiL(n.) o f class 1'.
As a n easy corollary of Theorem V , using (3.10), we have:
THEOREI\I
VII. If f ( x ) i s n functiort oj' 1111,n necessnvy a n d szcfficient
co~zditionf o ~the set o f i t s tr.anslntions to be closed dI1 i s t h a t f o r n o 21.
slzozcld zoe lznve
1
(3.25)
lW,l.(z)
W
e"'"dx
=0.
CHAPTER 11.
ASYMPTOTIC PROPERTIES OF AVERAGESZG.
4 . Averages of bounded functions. Our fu~idameiital tlieorein is:
THEOREM
VIII. L e t f ( x ) be n bounded nzeaszi~nblefitnction, defined over.
L e t K1(z) be n f i ~ n c t i o n in Ll , a n d let
( - )
f o ~every
?.en1 z c . L e t
lW
W
(4.02)
lim
Z+W
f(~)~l(~-x)d~=~J-:~<l(~)d~.
TIzen i f Kz ( x ) i s a n y f z ~ n c t i o n in L 1 ,
L,/.(F)
m
(4.03)
lim
X+W
K 2 ( 5 - x ) dF =
AS-:
lW( F )
Conce,sel!y, let Kl ( F ) be n f i ~ n c t i o n of Ll , a n d let
K 2 ( 6 )d..
W
Kl
clF
# 0.
Let
(4.02) i m p l y (4.03) whenever Kz (rc) belongs to L1 a n d f ( x ) i s bounded. T h e n
(4.01) holds.
26 The emphasis here placed on averages is in the same order of ideas as was first
introduced into the theory by Schmidt (1) (2).
26
N. W I E N E R . As to the first part of Theorein VIII, it is clear that it is valid whenever
2 i l k K, (x+ Lk).
lt=l
11
K,(x)is of the form
(4.04)
then
f(x)lsu;
aoO
I t is also clear that if
~-~i~~(x)--&(x)dx~r,
d6--lo0f
W
J-,f(6)
&(B
- J )
(:)Z;r(E-x) d ~ $
1 Be.
Ail application of Theorem I1 colnpletes the proof.
As to the second part of the proof, it is merely necessary to suppose that
1
Vs
J,h; (x)eitLIXdx
00 --
(4.05)
=
0
and to take
(4.06)
j'(n.)= e i u l ~
to obtain a contradiction.
Theorem VIII has an extension in the sense of Theorem 111, in which
Kl (x)is replaced by a class of functions 8 , (4.02)holds for every function
of the class, and there is no 21 for which for eceyy function K, (x) of 8 ,
The conveInse of this extended theorem is also valid.
5. Averages of bounded Stieltjes distributions. Here our theorem is:
THEOREM
IX. L e t f (x)be a fzt~zction of limited total vnricrtio?~o-cer every
finite raqzge, a n d let
(5.01) be boz~~zdedin y .
L e t Kl (x)be a conti~zz~ozrs
fzcnctiou o f L1, a n d let
(5.02) conz.er.ae.
Let
a r ~ dlet
(5.03)
S-mI{, (E
m
00
linl
X+W
- x ) df
(F)= A
T h e n if K, (x)is a n y f u n c t i o n o f XI,
-oO
Kt (F)d E
27
TAUBERIAN THEOREMS.
M
Q?zunsely, let (5.02) converge. Let Jpm K, (z) dx: # 0 . Let (5.03) imply
(5.04) for every fii~~ction
of & of illl nucl ecery f(.c) for zclAich (5.01) is
bounded. Then (4.0 1 ) Ibolds.
To prove this theorem we need the following elementary:
LEMMA
1%. Let
and let
With this lemma a t our disposal, the proof of Theorem I X does not
differ in any important respect from that of Theorem VITI. Of course,
as in Theorem VI, we must replace the polynomials in translations of K ,
that figure in Theorem VIII by absolutely convergent integrals in these
translations. The proof of the lemma itself is immediate.
The nlodification of Theorem I X with a hypothesis involving a rvhole
class of kernels will be used later, so we shall formulate i t as a separate
theorem :
THEOREMX. Let f (rx) be 11 fic?zct?ion of linlited total vaYiatio?l over every
Ji?zite ?.nnge, cuzd let
be bounded in. y. Let 2 be a class of continuous functions of L 1 , eaclt
one of which, for example Kl ( x ) , has the pro21erties that
converges, and that
S-M
00
(5.03)
lim
E+rn
Kl(F-r)df(F) =
AS
00
-M
Kl(E)dE.
Let there be no u which is a real zero for all the fztnctions
for. zchich Kl (x;) belongs to E. Tlzen ~f K 2( x ) is any f u n c t i o ~ belongin,q
to M I ,
28
N. WIENER.
lm
(F(F) d J
m
(5.04)
lim
x+m
m
IC2
X) d f
=
-m
K,(F)d F.
Conversely, if the class 2 contains at least one ficnction w i t h n o n zero
integral and has the property that eaclz member o f i t i s a member o f L1
a n d satisjes the condition t7~at(5.02)shall converge, and i f , twlzencver f (x)
satisjes tlre condition tlzat (5.01)be bozhnded, and (5.03)for every nzelnber
belongin,g to MI, then there
o f 2 , then (5.04)holds for e c w y fzcnction K2(x)
is n o 21 sztclz that (4.07)lzolcls for every fitnction Kl(xj belonging to 2 .
6. Averages of unilaterally bounded distributions and functions.
Let K,(x) not be equivalent to 0, and let it be a contiriuous function (or
a function continuous except for a finite number of finite jumps) not identically zero and such that
(6.01) K l ( x ) Z O ,
2
k=-m
max Kl(x)<p
(-m<z:m).
k:xzk+l
We shall sa.y that the mass-distribution determined by f (x) is boz~nded
belozu when
and that it is bozcnded above when -f(x) determines a mass-distribution
bounded below. W e shall prove:
LEMMA
X I a. L e t the distribution corresponding to f(x)be bozcnded below
(or above). Let (6.01)hold, and let
T h e n there is n Q s z ~ hthat
To prove this, let us notice that
Since K is "stiickweise stetig", B, a and b exist such t11a.t
Hence
(6.07)
M+N
2
k=-w
k:~:ktl
Kl(E) 2 ~ l z d J r ~ d f ( y ) ~ ,
TAUBERIAN THEOREMS.
or in other terms
2
(6.08)
~ l [ ~ ] + l } ( ~ + N
max K l ( 5 ) )
B
b-a
k=-cc kzxzk+l
2
If we combine this lemma with Theorem X, we get:
XI. Let f ( x ) be n Jknction o f limited total variation over every
THEOREM
jinite interval, for which
Let 2 be a class o f continziozts functions Kl, for each o f which
converges, and let
(5.03)
lim
x-+w
rcc
C
-m
30
K , (F -,z) d f (5)
=
A
-30
Kl ( 5 ) d 5
for every K, belonging to 2'. L e t there be n o rea,l u for which evely
vanislzes, for whiclz Kl belongs to 2 . Let Q ( x ) be a continuozcs function
belonging to 8,f o r which
T h e n if K2 (-x) is a n y f z ~ n c t i o n belonging to ill,,
W e shall have frequent occasion to use Theorems X and XI in a form
in which the infinite range of x is mapped on a semi-infinite range by an
exponential transformation. This may happen in two ways: either 0 or m
for the new argument may correspond to
m for x. I n the first case,
let us write:
+
and in the second,
(6.11)
5 = log A , f ( E ) =&-I
d p ( i ) , K I , I ( ~=
) iXj2(i),
411) = i d l ( L ) .
30
N. WIENER.
Then Theorem X I yields:
THEOHEM
XI'. L e t y (A) be a j i ~ ~ 2 c t i oofn limited total variatio?z ouer ece1.y
interval ( 8 , I / & ) lulzere 0 < ~ < 1 . L e t y(0) = 0 and let
Lct 2' he zc clnss of contirzuozts Jirnctions ATl(I) f o ~each of zuhicl~
2
(6.13)
pu -- -
cx,
inax A ~l~~
(I) 1
aI. 21:ak+'
co?zve~yes,ctnd let
for every
A\7,
( 2 ) beloizging to 2 . Let there be n o real t( for. which ew?y
~anisltes,for tulzich AT1 belongs to 2. Let M(I) be n f i ~ n c t i o nbe1ongin.q to 2 ,
for which
6 . 6
1
0,
if N2 (I) i s
Then
+Icx,
($1
4
(Iy p
I
! -
roost,. for 0 2 A < m
.
a129 continuozts firnction for zvlzich
(6.17)
max AIN2(L)I
li = -cx,
2k:122k+'
concer,qes,
I11
the case where
(6.19)
(6.12) mill be satisfied if
(6.20)
N
f (I) > - ---2log2 '
If the other hypotheses of Theorem XI' are satisfied, (6.18) will become
TAUBERIAN THEOREMS.
As a particular adnlissible N;,, we nlay take 1
AT2(p) =
(6.22)
{
, lo 2 ,I < 11
-1
1- -,
/I
I
.
[ l ~ ! ~ < l + f ]
.
[l+&i,lI Tl~us(6.21) will assume the form -
-
- Ns
41og2
'
Hence by (6.23)
--
(6.25)
Since
linl
11-+o(rn)
E
-
1.
f
is arbitrarily small, -
(6.25) (
A 1-
)
-
lim
i.+o:rn, +J
14'
+
;) 410gLL'
F
-
-
I1
,f(/r) d p
5 A.
Again, we may write (6.21) in the form (6.26)
lim
k-+0(rn)
[l
--L
(1+ 6)
A(l+E)
-r
f(/')dp]'=
4 1 4- ~ 1 2 )
I+&
from which we nlay conclude as above that
Combining (6.25) and (6.27), we see that
We thus get
T I ~ E O RXI1'.
E ~ I L e t f (.r) be n ,fic+2cfion bowzded over ecery i n t e ~ c l x l
( F , I/&), toliere 0 < F < 1 .
LP~ (6.29)
,f(),) > -- K (or ,f(A) < I<) nrgzlnzent. L e t S be u class o f ~ o n t i n z i o ~fii?~ctions
~s
hTl
(A)
enclz o f zohich (6.13) conce~*,qes,a nd let
, f i , ~every
f01'
32
N. WIENER.
f o r every A: (3") belouging to 2 . L e t the9.e be n o real 7 1 f o r ?luhiclz every
expression (6.15) vanislzes zuitlz ATl Belonging to 2 . L e t llif(A) belong t o 2 ,
and let
3
) 1
)
1
0
:i1
-
f(p)d p
5 const. f o r 0 < I -<
.
Tlzen (6.28) i s t m e .
Another case where a stricter conclusion than (6.18) may be drawn is
where y(A) is monotone. I n this case, (6.12) is automatically satisfied.
If we take & ( p ) as in (6.22), we shall be able to write (6.18) in the form
(6.32)
lim
[y (A)
A--to(m)
b (IS€)
+
d y (PI] = A (1 -k
(1 -
$1
and hence
(6.33)
Prom the analogue of (6.26) it follows that
and hence that
(6.36)
y ( I ) -An.
W e thus obtain
THEOREM
XI"'. If in the hypothesis o f Theorenz XI', (6.12) i s replaced
b y the condition tlzc~ty(A) i s monotone. T h e n (6.35) follozus.
It is even possible to weaken the hypothesis here given, and to replace
(6.12) by the condition that
(6.36)
lim
E++O
]im
-m<,u<m
[L
P U+C)
A-1
d y (A) i
J
L(~+E)
A
(
I
to establish (6.25).
CHAPTER 111.
TAUBERIAN THEOREMS AND THE CONVERGENCE OF SERIES AND INTEGRALS. 7. The Hardy-Littlewood condition. W e now enter upon the realm
of ideas which has longest been associated with Tauberian theorems.27
The class 2 of the last paragraph now consists of all functions of the form
?'
Hardy (2) : Hardy and Littlem~ood(4), (lo), (13) etc. ; Littlewood (1); Landau (9). (4) .
33
TAUBERI AN THEOREMS.
M (x)- M ( x f A)
(7.01)
for which we have
(7.02) (
-
= 1,
J-:
X(oo) = 0,
LV(E- x) df (g) bounded.
Condition (5.02) now asserts the convergence of
k=-m
+ A)1
max 1 M(X) - LW(X
k;x:k+l
whicli will be automatically fulfilled if M(x)is monotone and satisfies (7.02).
W e shall also suppose M(x) continuous. Condition (5.07)becomes the
condition that
m
1
(7.04) shall not vanish for any real 21 for every A.
oo , and let
exponentially a t
+
am
1
(7.05)
Let us suppose il1 to vanish
1
w (2)fFzcz x =
.
y (2)
y(z) will clearly be analytic over some vertical strip to the right of the
origin. W e shall obviously have
(7.06)
vI 2
--
sm
[llI(x)- 111 (X
+ A)]exZ dx = y
(2) (1 - ecAz)
-09
over this strip, and by analytic continuation, which we assume to be
possible, the non-vanishing of (7.04)becomes the non-vanishing of y (zj
over the imaginary axis.
Tf
(5.03)is satisfied for A
(7.08)
K
= 0.
I
) 4
&I($)-1
)
4
Then (5.04)assumes the form
which becomes
As is obviously permissible, let us put
)
F
;
(-oo<f'O)
.
(OsFie)
(8
E < m)
1
<
N. WIENER.
34
If w e put
(7.11)
(5.04) assumes t h e f o r m
(7.12)
f(-93)
LJ;+'
lim
=
0
f ( Z ) d Z = 13.
E
x+00
Restating T h e o r e m X I in t h i s n e w f o r m , w e get,
THEORE
XII.
I ~ L e t f ( x ) be a f u n c t i o ? ~o f limited total l;a?iation oz.e?. eve?-y
range ( - m , A), let f (-m) = 0 , a n d let
L e t M ( x ) be a monotonely dec~easingcontinuozrs ficnction, szrclz that
Let
(7.05)
analytically
over a strzp' to the ~ i g l ~o ft the origin, and let ~ ( z ) continzted
,
o?z to the i m a g i n a r y m i s , have n o ze).os there. ( W e assume t h e possibility
o f t h i s continuation.) L e t
(7.14)
be bounded,
iW(E - 2.) d f
alzc?
(8)
let
n(hl
lim
J
x+00
(7.12)
lirn
=B.
Y(B-x)df(Z)
-O0
-[ + E , f ( Z )
d Z = B.
6
x+00
L e t u s put iW(1ogx) = N ( x ) ;
j(1ogx)
=
F(x) T h e o r e m XI1 t h e n becomes :
T H E O R E MXIII. L e t F ( x ) be a ficnction vanishing a t the origin and of
limited total va?.iation ovey a n y finite range (includinq the origin), a n d let
f o r some A > 1 ,
L e t i i T ( z ) be n fnonotonely dec~ensirzgcontinz~ousfunction, szcc7~ that
(7.17)
Let
(
0
=
1,
AT(x) = O (x-9
Jim
I
VG
at
ac,
N(x) 2"-' cl x = yJ ( z )
[I, > 01.
35
TAUBERIAN THEOREMS.
over a strip to the right o f tlze origin, and let y ( a ) , conti)~z~ed
analytically
on to the ima,gi7zary axis, as we nsslinze to he possible for. all points bzct the
origin, have no zeros the9.e. Let
be Bouqzded, and let
JWivj'j ~ F ( s )
lim
(7.20)
=
x+m
B.
A case of particular interest and importance is where
rz1
F(x)
=
2 a,.
rr = 0
Here
(7.23)
Jy'
-
I : ' "
d ~ ( z1 -)- F ( n g ) 4- ~ ( y )
'
J
--
max 2 72 an ; (n
lo.
2
?L=[yl+l
(- n,, f a,
< [Ay]))
if A is sufficiently near to 1. (7.23) is t o be interpreted as meaning that
the expression on the left hand side is less than or equal to the greater
of the two expressions in brackets.
W e thus obtain the
COROLLARY.I f N ( x ) is subject to t l ~ econditions o f T l w o ~ e mX I I I , if
(7.24)
is bou~zded,i f
(7.25)
for all 72, a?zd i f
(7 -26)
tlzen
(7.27)
Jr
W
A'(f)d~($)
n n,,
<K
2
= n=O ilnx
(or
91 a,,
-K )
l
.
CCI
x+oo
1.1-0
W
2 a , -0
B.
Here instead of (7.27)) what we directly prove is
36
N. WIENER.
(7.27) may however be written in the form
liin
d5
an-= B.
10gAl
~L=O
5
Our conclusion (7.27) mill then follow if we can show that we can so
choose A,, that
1
lim z+oc
log A1
(7.29)
1
lim
m+m
log A1
However,
-
z+m
Again,
Since we may take A, as near to 1 as we wish, (7.29) is established, and
(7.27) follows a t once. This type of argument is to be found in the work
of SzAsz.
8. The Schmidt condition.28 This last corollary covers the work of
Hardy and Littlewood on Tauberian theorems. An extension of their conditions is due to Robert Schmidt. To arrive a t theorems of his type, let
M be subject to the conditions stated in the hypothesis to Theorem XII;
let us put
(8.01)
,c/ ( x ) =--=f (T a ) -- f(z)
and let us assume that
+
As the integrals in question converge absolutely by (8.02) and (7.13),
whenever they exist we may invert our order of integration and write:
-- -
p
p
-
- -
Schmidt (I), (2). Throughout this section, the author has been strongly influenced
by the methods of Vijnyaraghavnn and Szksz.
2d
37
T A U B E R I A N THEOREhIS.
Thus if ,11(2) is subject to the conditions in the hypothesis to Theorem XII,
if it is bounded above or below, and (7.07) is valid, and (7.14) bounded,
whether (6.02) is true or not, it follows from Theorem XI1 itself on replacing-
.f(.:)
-
by
+r
-w
g(8)dE that
Let u s now introduce Schmidt's notion of a "slo~vlydecreasing" function.
Let f (u) have the property that when 21 and 1: run through a sequence of
pairs of values for whicli
(8.05)
Zi 2 Z t ,
V-26--to
then
(8.06)
lim ( f ( v ) - , f ( z ~ ) )) 0 ;
ZL+w
we shall call f ( z b ) slozvly decrensit~g. (Schmidt treats sequences instead of
functions, and on another scale, but the difference is unimportant.) Schmidt
shows that we may write
(8.07)
f ( 2 6 ) =-,I; (21) -/--A
(26)
where f,(zl) is monotone increasing, and fi(.u) satisfies the condition that
whenever (8.05) is fulfilled, then
lim (fi(c) -fi (u))- 0 .
z4+w
He proves that there exists a function T ( a ) such that
lim T ( a ) -- 0
a-io
He proves that a number T exists, such that
'-
(8.11)
I fi ( v )-fi
(u)/ 2 T
(21 -26)
(v>
From this, it readily follows that if f is bounded near - m ,
valid and g ( z ) is bounded below. Furthermore,
ZL$
a).
(8.02) is
38
N. WIENER.
and
going to the limit
(8.14)
lim f ( x )2 - T ( 2 a ) + B
x+m and
(8.15)
-
lim f (z) 2 B-+TT2u).
x+m Then by (8.10), it follows that
(8.16)
lim f (
x+m x ) = B.
This yields
THEOREM
XITT. In the Injpothesis of Tlleorem X I I , condition (6.02) nxay
be yeplaced by the assumption that f ( x )is o f limited total variation over a n y
finite ran,ge, and is slowly decreasin,g, and the conclusion (7.12) m a y be
replaced by (8.16). Again, Theorem XI11 becomes
THEORE~I
X V . In tlze hypothesis to T h e o ~ e mX I I I , condition (7.16) m a y be
?.eplaced by tlie condition that i f zt and v rzcn th~ozrglt,n sequence for t ~ h i c h
(8.17)
then
(8.18)
v
~ 2 1 1 , --+I
2c
lim
- (F(v) -F
tl+m (u)) 2 0 .
In the conclztsion to Theorem X I I I , (7.21) m a y then be replaced by
linl F(x)
=
B.
x+oO
I n tlze corollary to Tlteorenz X I I I , (7.25) nluy be replaced by the Izypothesis
that if
(8.19)
S,L = no
4-. . C a,,
+
and p +x , q + y3 in suclz a. nzawzer that
then
(8.2 1)
lim (sq-sp) 2 0 .
P+W Tlze conclusio~z o f this corollary remains u?zcl~anged.
The hypothesis that (8.20) shall imply (8.21) is Schmidt's hypothesis
concerning slowly decreasing sequences in unaltered form.
TAUBERIAN THEOREMS.
39
It will be noted that the proof here given for the Schmidt theorems
does not essentially differ from the proofs given by Vijayaraghavan" and
Sz&sz30for theorems of these types. I n both cases the transition is made
to a Tauberian theorem of more standard type in which s, is itself averaged,
instead of appearing in the conclusion as the average of a unilaterally or
bilaterally bounded mass distribution. In both cases, moreover, the Schmidt
condition is again used to give a bilateral or unilateral estimate of the
difference between Sn and its average.
CHAPTER IV.
TAUBERIAN THEOREMS AND PRIME NUMBER THEORY.
9. Tauberian theorems and Lambert series. One of the most
important applications of Tauberian theorems is to the proof of the prime
number theorem of HadamardB1 and de la Vall6e Pou~sin,~"o the effect
that the number of primes less than N is asymptotically Xllog N. The
prime numbers bear an exceedingly close relation to series of the form
known as Lambert series, after their eighteenth century discoverer. Until
recent times, however, all attempts to employ Lambert series effectively
in the study of prime numbers had proved a failnre, and indeed Knopp3"
has characterized one of these directions of attack as "verfiihrerisch".
Hardy and LittlewoodB4 finally showed that the prime number theorem
was equivalent to a Tauberian theorem concerning Lambert series, but
did not succeed in establishing an autonomous proof of this theorem.
Our general Tauberian theorems suffice to furnish this autonomous proof,
and indeed, the Tauberian theorem which we shall find it easiest to establish
directly leads more directly to the prime number theorem than does the
theorem of Hardy and Littlewood. The latter is also directly demonstrable
by our methods. I n both, the cardinal point in the proof is that the
Riemann zeta function, 5(x $ iy), has no zeros on the line x = 1. The
proof of this goes back to the first proofs of the prime number theorem.
and has always been recognized as that property of the Riemann zeta
(I), (2). SzBsz (I), (2). 3' Hadamard (1). 32 de la Vallke Poussin (1). 33 Knopp (3). 31 Hardy and Littlewood (11) ; Hardy (6) ; Ananda-Rau (2). 29 Vijayaraghavan
30
40
N. W I E N E R .
function which is most central in the proofs of this theorem, but all earlier
proofs had made some use of the behavior of the zeta function a t infinity.
These further properties now appear as inessential, and the non-vanishing
of the function becomes the only non-elementary feature of the zeta
function in question.
For example, the usual proof of the prime number theorem employs
a lemma of Landau35 to the following effect:
LANDAU'S
LEMMA.Let
W
2 cc,, n-x
n=l
[R (x) > 11
F(x) =
(9.02)
and let
(9.03)
CAI,
20
[n
-
1, 2,
. . .I.
Let F ( x ) when analytically continued be without singula~itieson W (x) = 1,
except for a pole of order one at x = 1, with principal part Al(x - 1 ) .
Let there be some or for which
(9.04)
F ( x ) = O(lx/")
in the right half~lane. Then
In the following section, we shall show-following Ikehara- that condition (9.04) is not needed. When the lemma is used in the proof of the
prime number theorem,
a,,,= A,, (to be defined immediately) ;
(9.06)
F(x) =
-
5' ( x )/ 5 ( x )
and it will be seen that here too the only non-elementary property of the zeta
function which we use is that i t does not vanish on the line % ( x ) = 1.
To return to Lambert series, let A ( n ) be the number-theoretic function
determined by:
C ( p k ) = l o g y if p is a prime and k is a positive integer;
C(n) = 0
if n is not of the form pk.
=< x < 1; then
Let 0
W
m
W
W
x"
2 log nz xrn =nz=2 xm 2 A ( n ) =n=1
2 C ( n )nlnz
2 xn+ 2 C ( n )1-xn
(9.07)
1
33
Landau (1).
1
nlm
n=l
'
TAUBERIAN THEOREMS.
Hence
W
W
2 11 (7%)
=
1
2 log nL
-x?J"
Xl'l
1-x.
1
-
L$
1-x
x
log (14-
1
xnt
F [-b + .(;;.]]
- -
1-x
1
1
W
If we put x = e-5 and multiply by 5 we have
For E > 0, the derived series on both sides converge absolutely and uniformly,
and we may differentiate (9.09) term by term. On multiplication hg 5
this yields us
as 5-20. If we put
this leads us to
(9.12)
W
lim 5
S-to
2A
1
( 1 ~ ).ATl (9%
5)
= 1
.
This is a particular case of (6.14), which forms part of the hypothesis to
Theorem XI"'. A(n) is positive, as the hypothesis of that theorem further
demands. The function Nl (u) may be written
Thus of that part of the hypothesis of Theorem XI"' containing Nl, which
is now both M and the whole class 2 , it only remains far us to verify
the boundedness of
42
N. WIENER.
which follows from the fact that there is a K such that for 5 > 8,
>0
and from (9.12), and to verify further that
(9.1 6)
Now
(21) 2LiX
(9.17)
lim ( i x
==
d ZL
i+o
]in, ( i z
r?+o
= lim (i
+ I.)
+ A)
Jim
Jr
+ 1,) /'(is
u'x+'
cl I L
1 - eZ6
-k e-2t' + . . .) d l h
+ I -+ 1 ) (1 + 2- (i.z+d+l)+ 3- (ix+k+l) $ . . .)
d+o =
=
lim ( i x + A ) r ( i x + A + l ) C ( i x + I - t l ) d+o
+ 1) 5(ix -t 1 ) .
i x T(in:
Let us take over from the theory of the Riemann zeta function the
following facts:
(a) that the Riemann zeta function 5(s) is analytic on the line with real
part 1, except for a pole of the first order with principal part l / ( z - 1 ) ;
(b) that the Riemann zeta function has no zeros with real part 1. I t then follows that for any real
2,
and that
TAUBERIAN THEOREMS.
(9.1 9)
f
r 0
m
X1(a)dtr
=
;
lim i I . ( l b - 1 1 )f ( i + l )
R-tO
-
1.
Thus it follows from (6.35) that
Let us put z(11) for the number of primes less than or equal to
let us write
a(?&)= z ( u )
z ( d 2 ) +z(7b1l3) . . ..
(9.23)
+
+
++
I t is easy to prove by elementary means that
< cll'"[log
21Ilog dl
= 0 (211;"log 21).
(19.25)
-
+ 1)
Thus
We nlay write (9.22) in the form
la
1
h7
lim
(9.27)
,v-tm
log
t~ dm (71) =
On integrating by parts, we see that
(9.28)
lim
A~+W
Now
because of (9.26).
\
N
hi
Thus .
llrn
GJ
X+m
(9.31)
(AT) log AT -
N
-- x
-- ,v
m(N)
From (9.25) it follows that
z(N)
log N
log 1
Y
'
'
1
.
ZL,
and
44
N. WIENER.
This is the fainons prime number theorem of de la ValleB Poussin and Hadamard. 10. Ikehara's Theorem. The Landau Theorem (XVI) received several
successive generalizations a t the hands of Landau himself, and of Hardy
and Littlewood, perhaps the most general of which was indicated by Hardy
and L i t t l e ~ o o dto~ ~be the following: XVI. Let:
THEOREM
(i) the series Za,,
be absolutely convergent j b ~8 ( s ) :,-uO,,
0;
(ii) the fz~nction F(s) dcfi:fi.lzed by the series be re,qztlar for 8 ( s ) > c where
0 < c 5 o, itncl continuol1.s for 8(s) 2 c, except for CL sinzl~le pole zuitl~
l,esiclzte q at s = c ;
(iii)
F ( s ) -- 0 ( d l t )
(10.01)
fo). some finite CI, i~n.i~5irwz1y
fop*
(iv)
2
6 - c;
2 ,,
(10.02)
+l;
&-1
(v) a,, be j*eal, am? satisfy oue of tl~einequalities
or complez, and of tlze fo~wz
(10.04)
o { a;-'
Then
(lUlL
-
dlL= n, -t u2
(10.05)
}
+ . . + a,,
.
-
---1;
c
.
The vital change between the Landau and the Hardy-Littlewood theorem
is the looser form of (iii), which replaces a restriction of the form
Both these restrictions are inessential, and the true theorem is that of
Ikehara37, which reads as follows:
THEOREM
XVII. Let a ( z ) be a wtonoto~zeincreasirbg fitnction, and let
---
-- -
-
Hardy and Littlewood (8). 3i Ikehara (1). 36
TAUBERI AN THEOREMS.
con~ergefor !Ft (M)> 1
.
Let
convwge tcw$onnly to
it
finite linzit as
otiw any Jinite i?ztef.vulof the line !Ft(zr) = 1.
A
(10.09)
lim
.
K+oc
fl
Then
(N)
1v
a
To prove this, let us put
B(F)
(10.10)
5
= u ( e ~ ) e - ~. +
o J ~ z u ( e ~ ) d t . - A F . (5'0).
Thus
(10.11)
(l@(E) = e-:da($)--AdE.
Let us assume-what
is no essentJial restriction-that
TVllat we wish to prove is that
which is equivalent to (10.09).
If E >O and 7 is real, nre have
As this double integral is absolutely convergent, i t becomes
Jw
(10.16)
Sow,
e - ~ z d ~ ( gJ) n
-9)
- -lw
(i-
eitb'~-5)
dlt
-B
-"O
m
2(cosB(g--$)--I)
-B(T--F)~
-$,
e
dB($).
46
N. WIENER.
Thus
exist,s. Remembering that
and that
is a non-positive integrand, me have, by a theorem due to BraySs and
fundamental in the theory of the Stieltjes integral
and hence
-
i k ll
because 1-
,q (it1 $1) is summable over (- B , B). Similarly
may be proved to be bounded because of (10.23) and (10.12).
W e know that
(10.25)
47
TAUBERIAN THEOREMS.
and that all the other conditions of Theorem X I are satisfied, with the
possible exception of the non-vanishing of (6.15). To see that this is also
satisfied, we need only reflect that
Thus there are no zeros common to all these functions for all values of B,
and the non-vanishing of (6.15) follows.
It will be observed that the full force of our 7"~uberian method is
scarcely needed for this theorem. I n Theorem XI, which is the critical
part of the proof, the difficulty of proof is considerably lessened if K , ( x )
assumes such a special form as 2(cos BLT-1).
As a corollary of Theoren? XVII, wcL nlay prove:
T~IEOREAI
XTTIII. L e t y(x) be n mofiotone incrensi?zg fil?zctiot?, m21l l e f
convelye j o i , 8 ( ( 1 )
>1.
Let
(10.25)
( 1 )
-
[0 < B <
e f L ( 1 - 1)
101zen confinzted cii~alyficnlly,be ~eg-yzilcr?~
f o 8
~ (?r)
-,
1
1 nltd let it slot ~ , n s z i s l ~
To prove this, let us put
/ KO 5
log tly(:)
=
cc(2);
The theorem then reduces itself to Theorem XVII, provided we can establish
that (10.07) approaches a finite limit as B(u) +1. From the regularity
of (10.25), it follows that
(10.31)
y (u) A log (2( - 1)
+
is regular for %(u) = 1 , except for logarithmic sing~~larities
with ~ie,qc/atice
infinities. I t is also clear that there is no singularity for 11 = 1 .
Now.
l+o
LC
3 (y ("1)
so that
=
s-"(u)
cos (3( ~ 1 )log SC)c i y (2;)
48
N. WIENER.
+ + 2 i v)) = l+o
x-(~+') cos (2 v log x) d y (x) ;
00
W (y (1
(10.34)
E
and
+ r) ---- l+o
x-(l+" d y (x).
00
y (1
(10.35)
Thus
3 y ( l f - ~ ) - t 4W(y(1
$~+iv))+~(y(1+~+2iv))
(10.36)
.x-(]-+"' (3
+ 4 cos v log x + cos (2 v log 2)) d y (2)
and since
3+4cosy-1-cos2y = 3+4cosy+2cosey-1
= 2 (1 $ cos ~ p ) ~
(10.37)
20
it follows that
(10.38) 3 y ( 1 - t ~ ) - t 4 W ( y (+l ~ + i v ) ) + W ( y ( l 4-e-t-2iv)) 2 0 ,
or
(10.39) W ( y ( l S ~ - k ~ v '
) ) - l y ( l +~)-$W(y(1$-~+2iv)).
Thus
- S ( I P ( ~ +-E
+-~ ~ . ) )
lim
E+O
log 6
(10.40)
3A
W(y(1+~+2iv))
IP (1
&)
/ lim - -- - - $ lim
-4
5 - <4 I .
6+0
108~
E--fO
log E
+
\
On the other hand, if 1 $ i v is a logarithmic singularity of y with a
negative coefficient, it is a zero of F(u) of integral order ? I , and
- W(y(l+&+iv))
lim
log E
E+O
-
la
>I,
Thus gp (u) $ A log(u - 1) has no logarithmic singularities with negative coefficients, for W (u) = 1, and hence is analytic throughout this whole line.
Thus, by differentiation,
is analytic on the line in question, and the finiteness of the limit of (10.07)
is established.
I t will be observed that our proof, which completes the demonstration
of Theorem XVIII, follovrrs closely the lines of Landau's proofs9 that the
Riemann zeta function has no zeros on the line W (u) = 1, and includes it
39
Landau (2).
49
TAUBERI AN THEOREMS.
as a particular case. The prime r~uinber theorem itself is a particular
case of Theorem XVIII. Let us put.
(10.43)
Then
(10.44)
and the hypothesis of Theorem XVIII is manifestly satisfied.
becomes
(10.29) then
N
(9.27)
l = lim
N+W
M
.
logzda(x)
14-0
which we have shown to be equivalent to the prime number theorem.
Thus we have repeatedly shown that the prime number theorem is basically Tauberian in nature. I t might consequently be expected that the
more refined theorems as to t,he distribution of the primes, based on Rieinann's hypothesis as to the distribution of the zeros of the zeta function,
and established by Hardy, Littlewood and others, might be easy to establish on a Tauberian basis. Such a formula as
00
(10.45) lim 5
5+0
2 (A (n)-1) ?la
E-a
XI(12 E)) = 0
jO<a<l)
1
1
which follows from (9.10) inuch as does (9.12), appears to lend a certain
color to this view. Here the theorem to which this seems to lead is
and the condition of non-vanishing on the Fourier transform becomes the
Biemann condition
S(ix+ 1-n)$-0.
(10.47)
The author considers these hopes illusory and deceptive. Let it be noted
that (10.45) does not form a satisfactory hypothesis to a Tauberian theorem
until we have some hold on the boundedness of the mass distribution whose
integral is
Such information ~vouldalready presuppose as inucli information as (10.45)
can yield concerning all smaller values of a than occur in (10.46). Jn
other words, Tauberian theocems merely transform a 0 into a o .
1
N. WIENER.
50
Another way of stating the same thing is to say that a Tauberian
theorem always operates in the neighborhood of a single ordinate in the
plane of the zeta function. This is because it depends on a division of
the range of this function into near and remote parts, and because this
division has validity in the theory of functions of a real variable, not in
the theory of functions of a conlplex variable. On the other hand, the
more refined properties of the distribution of primes depend on the behavior
of the zeta function in the entire strip between ordinates 4 and 1 , inclusive,
and can only be discussed with the aid of Cauchy's theorem.
Of course, no proof of the limitations of so vague a thing as a method
has real mathemat,ical cogency. A t any time some super-Tauberian theorem
may come to light and prove to be central in the utmost refinements of
prime number theory. F o r the present, however, Tauberian theorems do
not seem to lie on the main avenue of progress.
CHAPTER V.
SPECIAL APPLICATIONS OF TAUBERIAN THEOREMS.
11. On the proof of special Tauberian theorems. I n the sequel,
we shall show that the greater part of all known Tauberian theorems
may be proved without great difficulty on the basis of the general theorems
of the present paper. However, most of the particular theorems were
proved in the first instance by entirely different methods. In individual
cases, these methods are simpler and more direct than the general method
here indicated. This is especially noticeable in the case of Karamata's40
proof of the original Abel-Tauber theorem.
Being in possession of a general method, we may consider with advantage
the particular methods and why they function. All Tauberian theorems
of the type discussed in this paper are intimately related to the solution
of an integral equation of the form
The most direct and general method of solving such an equation is by the
use of Fourier transforms. Nevertheless, there are many cases in which
a repeated differentiation will reduce such an equation to a linear
differential equation of finite order, and many more where the same repeated
differentiation will lead to a differential equation of infinite order, but of
manageable form. Thus it is appropriate in many cases to employ a technique
40
Karamata (2), (3), (4).
TAUBERIAN THEOREMS.
51
of repeated differentiation, and this has been done by Hardy, Littlewood4',
and Vijayaraghavan,4Vhough scarcely from an explicit consideration of
the integral equations in question. So far, the successes of this method
have been confined to cases where the analytic properties of the Fourier
transforin of F are extremely simple, and it has failed to throw any light
on the Tauberian theorems of prime number theory.
The methods of Robert SchmidtAglie more along the lines of the present
paper, in as much as he has seen the essential role played by the integral
i11 the study of the kernel of a Tauberian theorem. However, he has
devoted his attention to &(zt) for real integral arguments instead of for
complex general arguments. Furthermore, Schmidt's general theorem concerns the urlicity of the solution of l ~ i smoment problem rather than its
existence theory. As a consequence there is :L wide gap between his
general moment theorem and the particular Ta~tberiaiitheorems which he
obtains as corollaries. This gap he actually fills in in two cases, that of
the Abel-Tauber theorem and that of the Borel-Ta~tber theorem, but he
gives no general method by which it may be filled in in a new case. His
actual procedure is closely allied to that of Hardy, Littlewood and
Vijayaraghavan. Schmidt's chief service to the subject is in his great
improvement of ~vlrhatmay be called the auxiliary apparatus of the theory
of Tailberian theorems, through his invention of the notioils of "langsam
abfallende Fnnktionen" and "gestrahlte Matrizen".
Icaramata's elegant method leads to the study of the closure of a set
of translations of a given function, and thus most closely approximates
to that developed here. His function is
and the problem is solved through Weierstrass' theory of polynomial
approximation.
The translations considered are accordingly those of
the form
(11.04)
f ( x + log / I ) .
Sz&sz has carried further the study of this particular set of translations
of a given function. This is a far more difficult study than that of the
" CCf. Littlewood
(1).
(I), (2).
Schmidt (I), (2).
" Vijayaraghavan
43
N. WIENER.
52
closure of the complete set of translations, and here the general solution
of the closure problem is not yet known to me.
12. Examples of kernels for which Tauberian theorems hold.
Among the kernels admissible in tlie role of the AT(x)of Theorem XI11 are:
( 1 ) The Riesz kernels 44
( 2 ) The Abel kernel
(12.02)
(3) The kernel
hT(z) = e-x;
I
corresponding to the method of summation of the series Z a n , which gives
as its partial sum the Sbel average
(12.04)
of the cosine series
(12.05)
A Jm
( x ) e-AX d x
f ( x ) = zn, cos n x .
I n the respective cases we have
lW
N ( x )Y-l d x =
7c
2 sin
,
Z Z
None of these functions vanishes for purely imaginary values of z .
A possible 2\T1 ( p ) of Theorem XI"' is
r(A+ 2) I'(i2t
(12.10)
f 1)
r(it(+1+2)
.f-
0
if
I(
is real.
53
'IAUBERIAN THEOREMS.
13. A theorem of Ramanujan. As a lemma in our further work,
we shall find it convenient to introduce a theorem formulated by R a r n a ~ l u j a n ~ ~
and first proved by Hardy aild l ' i t c h m a r ~ h ,although
~~
with a formulation
somewhat different from that here given. The theorem is intrinsically
interesting, and is perhaps worth presenting in some detail.
Let f(x) be a function of L,, defined over (0, m). Then ed2f (e") will
also belong to L, over (- oc,m ) ; for
The Fourier transform of 8t2f ( F ) will be
The sine transform of f (x) will be
and its cosine transform,
g, (y) = V Z 1 . i. m.
7.c
B+oo
lB
cosxy d x .
Let us put
and
k2 (u) =
(13.06)
1
--- 1. i. m.
v2rr
We have
Ll'fi
z
(9) d y =
E+O
11"
g2 (y) yitL-112
d Y.
VzJoo
f (x) d x l zj : s i n x y
30 dx.
Similarly
(13.08)
45
sin xz
Lz ~ z g 2 ( y ) d y = V ~ 1 3 0 f ( x ) - - -x-z clx.
Hardy and Littlewood (8). and Titchmarsh (1). 46 Hardy
dy
54
N. WIENER.
However, on an exponential transformation, these become
]lzJ
E+c
w
(13.09)Jc
q (eTi2g2 ($)) d q =
e
-00
--
sin (eE+c) e
-00
*
(eu"(eF)) d 5 ,
and
If we now make a Fourier transformation and make use of the Parseval
theorem, it appears that
(13.11) kl (u)JW
(1- cos eE) e-6'2 eiui d 8,
e-7" e ' ~ ?d 7 = 0
and
(13.12) k, (u)JW
e-"
0
sin et e-El2 eiui d E .
dUUclg =
Thus
Here we make use of the formulae:
(13.131)
Y76
0
xv sin x d x = T ( Y f 1) cos 2
[--I
2 %(v)>-21
and
Y76
( 1 3 . 1 3 2 o) ~ 0 0 x v ( 1 - c o s x ) d x = ~ ( v + l ) s i n T
[-l>W(v)>-31.
Similarly
(13.14)
The duality of the relation between k, (u) or k2 (u) and h ( u ) is shown by
the familiar formulae of the gamma function,
TAUBERI AN THEOREMS.
Thus
(13.16)
and the real zeros of kl(u), k2(zc), and h(-26) are the same.
words, if the integrals in question exist, the zeros of
In other
(13
1
g2(y) y"-'I2 d Y 9
are the same. This is a very valnable way of determining new kernels
whose Fourier transforms (on a logarithmic scale) have no zeros.
14. The summation of trigonometrical developments. Let f (x)
be an even function of class L,, and, as above, let
fi (y) =
i. m. JBf(x)
% B+w
cosxg d x .
Let K(x) belong to L, over (0, a),and let us form
(13
(14.01)
This will eaual
x i g2(w)I<(tux) dw.
where
for the cosine transform of x K(tu x) is k
.
(3
-
Thus an average of g2(tu)
at infinity will appear formally as an average of z f(z) about the origin,
and an average of ~ , ( I u )about the origin will appear as an average of
1
z f (z) a t infinity. The kernels in the two cases will be K(x) and -- k (x).
x
Now by 5 13, we have formally
56
N. WIENER.
when 3 (A) = ;.
converge for 0
~f
5
L
- 4, i t then follo~vsthat
when, and only when,
Thus if k ( x ) / x in the role of n',( x ) , aiid A', ( x ) satisfy the other conditions
specified in the hypothesis of Theorem XI, and (14.07) holds, it is possible
t,o, infer from
S)
(14.08)
lim ~ J % ~ g ~ ( L u ) K ( , t c x=
) dAi u[
x+g
c 0
K(w)dlu
to the coiiclusion that
x
(14.09)
lim r L Wz f ( z ) N 2 ( ~ x ) d=
z d l X2(z)dz
z+g
The condition that f belongs to L2 may in nlaily cases be very considerably
altered and relaxed.
I t is possible to read the relation between (14.01) aiid (14.02) in the
reverse direction, and to infer from
and other conditions completing the hypothesis of Theorem XI, to
AL
rm
(14.11)
lim ~ ~ g ~ ( ~ ) K ~ ( =
~ u x j &
d ( ~z u t) ~i z c .
z-+o
The condition that
should determine a function over a strip to the right of the origin, which,
when continued has no zeros over the imaginary axis, becomes the same
condition for
57
TAUBERIAN THEOREMS.
We thus arrive a t a whole class of theorems relating the partial sum of
a Fourier development with the average of the funct.ion represented about
some point. F o r example, it is easy to prove:
THEOREM
XVIII. If f (x) i s a non-ne.qative fzilzction of class L, over
(-rr, x ) , if i t i s sz~tnnzable at a n y point 1)y a Riesz nzean o f a n y positive
odes., or by a n Abel nzean, to a n y value A, it i s so sumnzable to A by
Riesz means o f all positive orders, and by a n Abel mean. A necessary and
acfjcient condition that this should take place at a given point x i s tlzat
This theorem is due to Hardy and Littlewood.
15. Young's criterion for the convergence of a Fourier series.
\Ire now come to a region in which S. B. Littauel. has done work, which
constitutes the theorems proved in the present section. Young has proved
the follo~vingtheorem: for the Fourier series of tlze integrableficnctionvf(u)
to converge to s f o r zc = x, i t i s szcfjcient tlzat
(15.01)
and that
for small t , zulzel*e
Further work on this theorem has been done by Young, Pollard, Hardy
and Littlewood. The chief theorem to which Hardy and Littlewood came
was the following:
THEOREMXIX. d necessar'y nncl swfficient condition that tlze Fourier
series o f the integrable fiinction f (u) be sz~mmable (C, - 1 6 ) f o ~a n y
positive 6 i s that
(15.04)
Yl(t) = 0 ( t )
r
provided that
+
1
This theorem in its original form is not directly adaptable to proof by
the methods of this paper, inasmuch as Ceskro summation of fractional
58 N.
WIENER.
order does not depend on a kernel of the form n'(nz). On the other
hand, if the Cesaro summation is replaced by Riesz summation, the theorem
is reduced to a particular case of theorems already proved. W e shall
here content ourselves v i t h proving this related theorem, as applied to
Fourier integrals rather than series, in the case where f ( x ) differs from
zero only over a finite range.47 Let us write
t IP(t) = Y(t).
(15.07)
Then (15.05) becomes
which yields the bonndedness of
(15.09)
Furthermore
Thus (15.04) becomes
(15.11)
W e have
This is similar in form to (6.15).
The Eiesz sum of k t h order of the Fourier series of f ( x ) - s for x = 0 is
m
-
(15.13) lim
o
.
1
It r (t1 - l-- k ) . f t
d 7 (1,)
I" t
(1 - i)kcos -dl.
8
m
= Sl+ Oi
---
-
-
dt
m L ~ m c i p ( t ~ -)t r-~( -l +~ k~j ( ~ - ~ j ~ ~ 8c o s ~ ~ ~ ~ w
-
*'It is possible to make the transition to Theorem XM, but several of the steps require some consideration. In particular, the equivalence of Riesz and Cesiro summation for
orders 1 - 1 was proved by Riesz (Proc. Lond. Math. Soc., 2 (22), 412-419 (1924).
TAUBERIAN THEOREMS.
Here if
16
> - 1,
is finite, and the illversioil of iiltegratioii in (15.13) is hence applied to an
absolutely coiirergent integral. i n view of (15.08).
TVe have
1
00
r(l
+ k)
1=1
1
Lm
J" (1 + 1) ~ ( k )
1
- J1
(1 d 2 LLli
(it4 + 1) T ( k )
=1 -A ) 2 - A
(iu+1 ) T(k)
-
,.,,iu-l
(izc
I ~ ) J C - ~
).)h-l
(15.15)
W
A-0
sin 2 w A
wit&--1sin
1.W d w
1
r(k)
(iu+l)T(k) T(k+l-izc)
rn
-9
T ( i t ( )cos (iu- 1 ) 2
--
-
T ( 1 - i u ) T(iz6)
(izi+ l ) T ( k + l - i u )
.
ziu
sin -2
zi u
.-
-
n sin -2
(izc+l)T(k+ 1-iz~)sin,zizc
In proving this, we have made use of (13.131) and (13.132).
The other conditions of the hypothesis of Theorem XI' are readily
proved to be satisfied, and we have already show11 that if (15.05) is satisfied,
(15.11) and
60
N. WIENER.
w e eqztivnle?zt. T h u s g ( 1 5 . 0 5 ) is scltisjied, (15.04) i s completely equivalent
to the stcitement that the Fou?.iel- s e ~ i e so f f ( x ) i s su~7znzuble to s by Riesz
(or C e s d ~ o )szinzs o f a n y gizsen o?-cler eccccedir~y 1 .
16. Tauberian theorems and asymptotic series. Certain asymptotic problems arise in the discussion of the behavior of an integral
for large values of z. By a change of variable, this problem may be
reduced to the consideration of
p
(16.02)
e-& @ ( t ) d f .
In particular, let ns discuss the situation which arises when zJ>O and
which we nlay write
is bounded and O ( t ) is positive, it will follow by Theorem XI" that
or that
Again, it will follow that
which yields
(16.09)
linl
E-+O
The condition that
(16.10)
lD (+)
tY-10
A
r(v)
dt =-
r ( i u + v ) = L m e - l tiU+v-ldt
'
#0
61
TAUBERI AN THEOREMS.
is obviously satisfied. Indeed the Tauberia,n theorems of this section
(which are due to Ikehara4') differ only from those of the last section in
that x tends to infinity instead of to 0.
If
2
is bounded and tends to B a t the origin and v > 0 ,
Then
W
(16.12) lim x
2300
.Bxvl
30
l (xt)z'-lo-xtodt
tv-1
Thus
linl
(16.13)
X+W
r'r
e-"
=
~9(t)
tv-le-xtdt
=BI'(v).
d t = R T(v).
In particular, let there be a neighborhood of the origin in which @(t) is
v- 1 times differentiable, with a bounded derivative of the (v -1)st
order. Let
(16.14) @ (0) = 0 ; @' (0) = 0 ; . . . @(tf-2) (0) = 0; @(lf-1)(O) = A
0 (t)
and let 7-7
be bounded outside the neighborhood in quesbion. Then
(16.03)
lW lim rv
x+X,
ecxt @ (t) d t = A.
If now 0 (t)- Atz'-l has a bounded derivative of order y -1> v - 1 at
the origin, and vanishes there with all its derivatives of order less than
A tV-l
p - 1, and W(t) is a function which equals 0 ( t )- ----- in some neighr (v)
borhood of the origin, and for which
2
is bounded and tends to A a t
the origin, we have
(16.15)
lim xp p r e - ~ ~ u i ( ~ ,=
l t A.
x-+m
By a repetition of this process, it is easy to show that if
if W(t) is bounded and ([c - 1) times differentiable with bounded (p- l)st*
derivative in some neighborhood of the origin, if
49
In n paper not yet published.
62
N.
and if
y
> a?, . ~s > 1 1 , .
. . .,
WIENER.
> vl;,
then
as x + m . This is an adequate basis for the theory of asynptotic expansions of Lnplace integrals, and enables TVintner's workd9 on the subject
to be greatly simplified.
CHAPTER VI. KERNELS ALMOST OF THE CLOSED CYCLE. 17. The reduction of kernels almost of the closed cycle to
kernels of the closed cycle. In the present section, we shall approach
very close to the work of Robert Schmidtso in his discussion of "gestrahlte
Mittelbildtulgen", although our terminology mill be somewhat different.
Up to this point, we have been discussing means of t,he form
(17.01)
J-m
Kl (2 - s ) f (F) d 5
or means only differing from these by a change of variable from 5 to
a function of 5. Let us now turn our attention to means of the form
where
(17.04)
Kl(t"! zj
= ~ , ( ; -ix ) +
K:(E,
x>
where K,(x) satisfies the condition that (5.02) converges and that (5.07)
does not vanish, and where
00
(17 -05)
lim
x-+m
2 n zmax
IK?(~,
?~zn+l
-m
x)/
Let K,(x, y) and Kl (2) be continuous, and let
49
Wintner (1).
Schmidt (2).
=
0.
TAUBERI AN THEOREMS.
be bounded.
Let
==AS
m
50
(17.061)
.T
lim- + S-OOK1(E,x)dr(E)
~
-m
Kl(5)dl.
The argument of Lemma X I a will need no substantial alteration to show
that there is a Q such that
I t will then follow from (17.05) that
and that
n
C
A
is bounded.
Hence, by Theorem X, if K, is any fuilction belonging to Nl,
In Chapter 111, the kernel K (5 - x ) is replaced by a kernel of the form
where 1W(z) is a monotone function for which
and for which the function
(7.05)
which is defined over a strip to the right of the origin, when continued
analytically on to the imaginary axis, has no zeros there. W e can replace
the kernel M (5 - X) by a kernel of the form
where Ml(x) satisfies the conditions we have already laid down for M(z3,
and where
(17.09)
and
(17.10)
lim
x+00 1
I dill2 (F, z ) I = 0
iM, (F, x) = O (e-242) at ao uniformly.
Under this change, Theorem XIV still remains valid.
64
N. WIENER.
18. A Tauberian theorem of Hardy and L i t t l e w o ~ d . ~
Hardy
~
and Littlewood have proved the following theorem:
X X. Let f (XI
a n z n be a power series with positive coefficients,
THEOREM
and let
A
(18.01)
f ( x ) (1 - x)" (a>O, x - t l ) .
Then
78
n" A
( 1 8.02) Cnk(n-a).
r(a+l)
1
W e may write (18.01)
a,$e-nt
A $-a
(18.03) or
=z
-
2
e-"E = A .
2n ~ - Al (n
lim 5
( 1 8.04)
E-to
This falls under (6.30), and
(18.05) Since a,, Inu-l
lim 5 F
5-t
Eiu ,j
=
r (a+izc) =/= 0 .
>0 ,
11/51
(18.06)
e-E
$a-1
a,z (12
5)"-1 JoP
-
0
A
A
---~ " - 1 e-i c i ~
r(a+1
rW
r O
1d8
because of Theorem XI"'. Writing tn for 118, and letting m become infinite
through integral values, we get
which is only another way of writing (18.02).
This however is not the most general theorem proved by Hardy and
Littlewood in this connection. They show that if
(18.08) where
(18.09)
and
(18.10)
then
j'
f (x)
-
(1
A
qcT)
L (u)= (log w ) a l (log log Z
Hardy and Littlewood (4).
11.
1
!
. . (log(lL)zt)")l
L ) ~ ~
L (u)# 0 (1) a t oo ,
65
TAUBERIAN THEOREMS.
We may write our theorem to be proved in the form that if
then (18.11) follows.
This we may again write
as p + m . Now, if M ( $ ) is a continuous function, defined over (0, oo),
and asymptotic to L(S) a t oo, we may show that:
Here M(E) is introduced instead of L(E) in order that we may have no
trouble with finite singularities of L (5).
To prove (18.14), let us reflect that
+l::
e(tb -@
(Y
e-e(il-P)
M(eU)d
~ t
Since M ( $ ) is asymptotic to L(S), which is asymptotically increasing, the
first integral is asymptotically less than
As to the second integral, i t is asymptotically
while the third one is ultin~atelyless than
66
N. WIENER.
Now.
C(,-312
--.
(18.145)
00
5 e--14
+J
"I-':~
--
< 2 e-ILL
Thus by (18.144)
(18.146) J'O
e(tl-,l~)C4
LLT li
p-t?("
'"1
.
p"l-l12
M(eU) 117 t t
5
( a , - f) *La'-'"
d zt
j.
iU
eA2e-eA [L (ep)
+ o (I;(efl))].
Combining this with (18.141), (18.142) and (18.143), we see that
where y ( d ) vanishes as 9 becomes infinite.
(1 8.14) follows.
Consequently, by (18.14), if we put
Since A is arbitrarily large.
On the other hand. we may write (18.11) in the forin
Both integrals, (18.16) and (18.17), converge absolutely. TVe wish to make
tlie transition from (18.16) to (18.17).
If a , 5 1, ?,b (21;) is "slowly decreasing" in the sense of Schmidt, we
can apply Theorem XV, and it appears that (1") + 0 as 1. +cx; . From
this it follows that
TJTe now come to the more general case where a, > 1 . Our hypothesis
becomes
log 12[""
e(~ogu-,u)( c e - l ) c,-e lovl&-,u
a (18.19) e - , ~ - -2
AA(ePj,
1lC4-l(log n)[("]
I"
2
-
TAUBERIAN THEOREMS.
where
(18.20)
-4(16) = (log 26)a,-[all (log log 26)""
The kernel of (18.19), in the sense of
5
. . (log"" 26)"$'.
17, is
I t hence appears that this kernel is of the form indicated in (17.04) where
K, and KT satisfy the appropriate conditions. Thus if we put ill*( 5 )- A ( E )
at oo and M" (8) is continuous, and if
it follows just as in the case where
or, _<
- 1
that
This however is only another way of writing (15.11).
19. T h e T a u b e r i a n t h e o r e m of B o r e 1 s u m r n a t i ~ n . ~ V hBore1
e
sum of tlic series with partial sums s,,, is
Let us put
12
=
[ti2]
and
ior 11 > O .
Let f(u) = 0 for negative
Then (19.01) becomes
(19.03)
lim
?/--too
r
c'-vP
ti.
Let us also put x
Pi:
dJ'(u)
?, - V?12- 1
I'i2h2+l)
Now
For this section: cf. Schmidt (2) ; Vijayaraghavan (1).
jZ
-
-
s
=
y2,
68
N. WIENER.
Thus over the range y - yll"
zu
because
5 y+
y1j6,
):(
2uP
e8t~'-4u~+~s
Moreover, it is always true that
Thus the conditions (17.05) and (17.06) are satisfied if we put
and
(19.09) I t follows a t once t)hat (19.10)
and that
YFJI
lim
Y-+W
(19.101)
e-2
(21-yj2
e-2 (u-9)'
df( U j = s
f( 2 k )
is bounded.
W e now introdnce again in its appropriate form Robert Schmidt's
definition of a "langsam abfallende F'olge". The sequence {s,) has this
property if, whenever q = q(p) (p = 0, 1, . . .) runs through such a sequence
of indices that
(19.11)
q
2
and
q -P -+ O
v23
TAUBERIAN THEOREMS.
then
(19.12)
Schmidt proves by elementary methods that if so, sl,
then
C(A) =
lim
(s,- s,)
(19.13)
..
is such a sequence,
P--too
P:q:P+ifVF
exists, and
(19.14) lim C(A) = 0.
A+o
Furthermore, he shows that there exists a constant K, such that
for all p = 0, 1 , ... and all p
exists such that
2 p+ VF.
SP
(19.16) Vp+l
2 -K
Thus he shotvs that a K
( p = 0 , 1, ...).
-
Still
Let the sequence s, be "langsam abfallend" and let (19.01) hold.
following Schmidt, we see that
If we build up the appropriate f ( u ) as in (19.02), and make the transformation which led from (19.01) to (19.03), we get:
Again, by (19.07)
(19.19)
Obviously
(19.20)
e-~zv n x n
o
n!
.
e-(tb-vy)z 0 (u) d u = 0 ( K T )
70
N. WIENER.
On the other hand
L Z T ~ e~-x~xI
e-x X: x
)&
2 -r> lim
zzm
n,
[XILl
,r+m
(19.21)
=
lini
e-x-l
x
(x+
cl
V F )V ~~Z ( X + V F )
I;+(
[ I/-I
112
=
--
Thus (19.17) yields
(19.22)
I& x-ll%srxl<
r+m
+
[ ~ x - l ' ~ I<0 (1)
+ C(1) cc-lr]
=
0(I),
which we may simplify and write
Let us now introduce the function
Substituting this in (19.25), and ~vritingS for an average of the sn's for
xY<z<(~+2af
) ~1, we have
TAUBERIAN THEOREMS.
I t is indeed easy to show that as z increases, the weight of the later
s,,'s averaged in S increases a t the expense of the weight of the earlier ones.
We niny show that the condition that { s m } is "langsam abfallend" may
be put iu the form that if
(19.28)
then
lim (srpl-S I , ~ , ) 0.
(19.29)
-
u+00
Hence by (19.27) and (19.23), we may readily shorn that
- ( y u ( ~-ga
j
lim
(70) 2 0.
u+m
Again remembering (19.23), which makes all the integrals in question
absolutely convergent, we have
An integration by parts yields:
72
N. WIENER.
Similar methods establish the boundedness of
We may then apply Theorem XIV, taking
In this case
(19.35)
Hence
(19.36)
lim ga(x) = s -
x 3 w
Let us now return to (19.27), making use of the existence of (19.13)
and of (19.14). Then if x is sufficiently large,
(19.37)
and
(19.38)
It then follows from (19.27) and (19.36) that
-
lim s,,
5
s - C(5 a)
)1+00
and
(19.40)
lim
- s,
n+oo
2 s + C(5a).
<=
Combining these, and remembering that C(A) 0,we see that the following
theorem holds :
THEOREM
XXI. Let (19.01) lzold, and let s, be "lautgsam abfa12endv.
Then
(19.41)
lim 8% = S .
?E 3
CHAPTER VD.
A QUASI-TAUBERIAN THEOREbI.
20. The quasi-Tauberian theorem. Up to the present point, all
the Tauberian theorenis we have discussed have involved some auxiliary
condition of boundedness or positiveness or slow decrease. In the present
chapter, we shall discuss a theorem without any such auxiliary condition
as to the function averaged. The type of theorem is so fundamentally
73
TAUBERIAN THEOREMS.
different from that already discussed that the nomenclature, ('Taubwian",
seems to the author unfortunate. The theorems now to be discussed are
in essence much closer to Abel's theorem than to Tauber's theorem.
By an integration by parts, me establish the following:
LEMMA
XXII a. Let
exist for every C > B, and let
exist as the limit of
CF(X)
d f ( x ) ns C'+ .a. Let
exist for. each C> B, let G ( x ) and F ( z ) be continztous, and let
Then
(20.05)
will exist, ancl
(20.06)
- LL'
d f ( x )- r
rW (%)
B
d
L F ( $ )d f ( 8 ) .
Now let us suppose that f ( x ) is of limited total variation orer every
finite interval, that Ii, ( x ) is bounded and continuous, that
(20.061)
'
d
( x )e43 i const.,
and that
(20.07)
exists in the sense of (20.02). Let us suppose further that as x
(20.08)
where A1 .f. 0 . Then
Kl ( x )-- A1 e'",
(A > 0 )
+ -m
74 N. WIENER.
/ < const.,
<
if D is sufficiently large that for cr: - -D,
/ e-j.~ li; ( z )i 2 const. > 0 .
(20.082)
As a particular case of (20.06), if B is large enough,
(20.083)
JZm- 9d.r.w
and hence
(20.084) is bounded. A further application o f (20.06) yields us Froill this we may conclude that
(20.09)
O(1) a s y + - - c c
and that
lini e-lv
(20.10) C-+m
K, (
-2 )
f (
=
0
uniformly in y.
iVow let R(z) be a fonction for ~vliicli
and
(20.12) are finite. I t follows from (20.10) that ,
~ ; L Z R ( ~ - Z ) ~
=
(20.13) Kl(z-x)d f (x)
linl J - ) B ( ~ - ~ ) ~ K J ~f(xl
- ~ ) ~
B-,m
= B+W
lim
~~df(4~S",li,(z-z))d~(y-z)
=i r n d
(z)S-", ~ ~ ( y - - z - z ) dB e ) .
TAUBERIAN THEOREMS.
Thus if
m
(20.14)
71 lirn
-+
r ~ , ( ~ - x ) d f ( d= d l M l i l ( z ) c ) d e
then
$=K,(zJ - 2 j d
(20.15)
(2)
will be bounded: by (20.09), and it follows that
lam
0%
u'
Hewe
k; (x) i s boz~nded and co?ltinuous, if j'(z) is oj' limited total
variation eve,. cuevy Jinite interval, i f (20.14), (20.061) and (20.08) hold.
iJ' (20.11) and (20.12) are finite, and if'
l h l.s is a sufficiently important theorem to dignify by a numbel-; we shall
call it Theorern XXII.
A closely related proposition is:
THEOREM
XXIII. In tlze Izypothesis o f Tl~eovenaS X I I , in case
11
(20.19)
K, (z)
= 0,
[x> 01,
(tie nzay 1.eplace the asszinzption o f t l ~ eJinifeness qf' (20.11) a ~ ~ (20.12)
r?
by
that o f tlze Jiniteness oJ
(20.20)
T h e conclusion remains valid.
76
N. W I E N E R .
To prove this, let us reflect that if (20.08) holds as x + m , since
it will follow from (20.06) that
By a precisely similar argument,
a*
if B > y. Thus by (20.21), JB Kl ( y - x ) d f ( x ) is less than a function
which is bounded and decreases monotonely to 0 as B becomes infinite.
Consequently, by a theorem of Daniell on the Stieltjes integral,53 in combination with the finiteness of (20.21),
Thus the inversions of integration in (20.13) are again permissible.
Formally and heuristically, (20.17) is equivalent to
(20.25) Lrn
K , ( x )eUxd x
J-:
Lw
?o
-- =
K, ( x )eUx d x
If Kl and K, are both O(e-(p+E)x) a t
etmdR(x)).
+ cx, and O(e+€")
a t - cx,,
Lrn 00
(20.26) and
h2(u)=
(20.27) k1 (u) =
53
Daniell (I).
K2( x ) eUx d x
00
Kl ( x ) em d x
TAUBERI AN THEOREMS.
are both analytic over - E
< % (u)< E + p .
Formally,
the second integral being taken along any ordinate in the strip
-E<%(~l)<E$,lb.
Nozu let Kl (x) belong to L, Let k, (zc)lk, (u) be analytic over -- E
% (u)
5 p E , and let i t be quadratically sz~mmableover every ovdinnie in that
strip.
- Then
.
+
(20.29)
and
are gzcadratically summable.
As a consequence,
(20.31)
and
are absolutely summable. If we assume R to be defined as in (20.28)
and p = 1, we obtain:
THEOREM
XXII'. Let Kl ( x ) be bounded and continzcous. Let j'(x) be o f
limited total variation over every finite i n t e ~ v a l . Let (20.14), (20.061) and
(20.08) hold. Let ke ( u ) and kl ( u ) be defined as in (20.26) and (20.27),
respectively. Let Kl ( x ) belon.9 to L,. Let k, (u),'kl (u) be analytic over
- E 5 % ( u ) 5 I E , and let i t belong to L2 over every ot~dinatein that
strip. Then (20.18) follolus.
Similarly, we have :
THEOREM
XXIII'. In the lzypotlzesis of Tlzeorenz XXII', i n case Kl ( x )
I % (u)51 E
vanishes for positice argunzents, zue m a y repltrce the strij -F by the narrower strip -E 2 % (zc) l s .
21. Applications of the quasi-?'auberian theorem. If
+
+
78
N. W I E N E R .
then (80.08) is clearly satisfied for 1.
= 1,
and
K, ( x ) = Kcm)
(2)
K, (z)
If we pnt
(21.05)
z
,
-
(20
=
(""'K(x)
. K("L)
(x)el&"
-=
+
+
r(rn 2) r ( u 1)
r(?h+m$2)
'
and
then as
13(N) + K
-
2r(nz
+ 1)
--
Thus if n > rn, then
lim
(2 1.08)
will imply
iirn
( 21 .09)
Jo
(?,OK(y - .r) d.f ( x )
Y4m
y--7m
[
K(70(y
-
1 3 ( I t ] ~#(t6)-71&+9?+1/2,
d Jo (Ib)K
(x)d J30
-
Kc")( 2 )d T .
z ) d,f ( 2 ) = d
+
-A
while if nz >, ?z 1 , (21.09) mill imply (21.08).
With a little manipulation, already indicated by the authors in question,
this result is seen to be equivalent to an important theorem of Hardy and
L i t t l e w ~ o d , ~ h h i cgives
h
the necessary and sufficient condition for the
summability of Fomier series and integrals by Ceskro sums of some order.
In the integral form, the theorem reads as follows: Let j'(x) be n nzea.mrable
,ficlzcfion definer1 over (- G O ,GO),and zero outside ( - A : A). L e t
Then i f
lue
zc~rite B, <for tlze propositiolz
ant1 Ti,,, ,fo,. the p ~ o p o s i t i o ? ~
Bnbinzylies
j1
Cnt+e
, ~ o I171.. 2 1 , 11.1iile(/na inz~~lies
B m + l i-z .for nz > 0 .
-
Hardy and Littlewood (13).
TAUBERIAN THEOREMS.
79
lT7hile the general method of the present section has been developed in
an earlier paper of the author,55 his final results mere not stated correctly.
~ ~ Paley, who have shown it
The correct result is due to B o ~ a n q u e tand
to be a "best possible" result in both directions. Their theorem also
applies to m > - 1, and is otherwise somewhat more general.
CHAPTER VIII.
TAUBERIAW THEOREMS AND SPECTRA.
22. A further type of asymptotic behavior. The 0 and o symbols
do not exhaust the possible terms in which me may describe the behavior
of a functioil a t infinity. 4 proposition which nlay be regarded as in
some vise a generalization of
lim f ( x )
z+m
+
1
,-J
lim
O-im
=
A
PB
, f ( x ) - ~ ~ -~0a ~
U
If c f ( x )is bounded and measurable, (22.01) clearly implies (22.02), while
(22.02) does not imply (22.01). Proposition (22.02) has a certain analogy
to the different types of "strong convergence" to a limit which a function
may exhibit: namely
%
M
(22.03)
and
f(x)- d d r converges
1 00
(22.04)
f( T )-A d x converges.
1%
The series analogues of the latter
(22.05)
5
5!
s,, - - A
converges
1
and
(22.06)
s,, -A ' converges
imply the ordinary convergence of s,, to A: but are not implied by it.
On the other hand, neither (22.03) nor (22.04) is implied by ordinary convergence, although (22.04) implies (22.02). 111 cont,rast ivith propositions
Wiener (5). "Bosanquet (1); (2): Paleg (1). "5
80
N. WIENER.
(22.03)-(22.06), which represent various types of "strong convergence",
we shall express (22.02) in the usual language by saying that f(x) is
strongly summableb7 to A as x + m . We shall say t#hat A is a sztblimit
of f (x), and shall write
(22.07)
slm f (x) = A .
z-+m
The sublimit of a function f (x) differs markedly from the ordinary limit
in that neither its existence nor its value are invariant if we replace x
by a monotone function of x becoming infinite with x . On the other hand,
the sublimit of f(x) has a relation to the harmonic analysis of f(x) far
closer than does the ordinary limit.
Closely related to the notion of sublimit is that of subboundedness, which
bears to the ordinary notion of boundedness much the same relation which
that of sublimit does to the ordinary notion of limit. A function f(x)
is said to be subbozc.nded if
is bounded. This juxtaposition of a notion of limit and a notion of
boundedness suggests a generalized form of Tauberian theorem. To be
specific, let us ask what conditions beyond the subboundedness of f(x) are
sufficient to make
00
M
(22.09)
slm
K,(x--5)f(E)d5
=
IL~K,(F)~E
,Z+M
imply
(22.10)
slm l " K 2 ( x - - E ) f ( E ) d ~ = A
.Z+W
This problem belongs to the range of ideas treated by the author in his
work on generalized harmonic analysis. It gives a clearer picture of the
real significance of Tauberian theorems. On the other hand, it does not
a t present offer an alternative approach to them, since Tauberian theorems of the type already discussed in this paper play an essential role
in the establishment of a theory of generalized harmonic analysis. Results
involving these theorems will be applied in this section.
To return to (22.09) and (22.10), no generality is lost by taking
A = 0, as this simply amounts to replacing f (x) by f (x)- A . Let us
then take A = 0 , and let us put f(x) = 0 for negative arguments, which
j7
Hardy and Littlewood (15).
81
TAUBERIAN T H E O R E M S . is cleal-ly perinissible. IVe then wish to find a set of conditions which
in conjunction with the boundedness for large B of
(or what is the same, with the fact that
f(x)
- belongs
14- lz: to L,) and the
proposition
are sufficient to imply
We shall prore the following: THEOREM
XXIV. L e t f(x) be menszrrable, a n d let (22.11) be bozrndelt.
L e t (22.12) be calid. L e t K, a n d Ar2 be measzcrable, a n d let
(22.11) fr ( l + l z ~ )I K~~ ~( X ) ,d~2 <
m
e
-w
(and henceJ-:
and
(I
+ z 1)
rm
(l+ XI)"'
(22.16)
/ Kl (z) )
d x <a),
IKx(~))'dx<m (and l ~ e n e e (r l~+ x l ) JK,(z) d z < a ) . F t ~ r t h e ~ n z o r c(et
~,
for all veal z t . Z'lzen (22.13) i s t ~ z i e .
F o r this theorem we shall need the following: LEMMA
X XIVa. L e t f (3, u,1) belong to L2 as a ficnction o f
LmIf
am
tixea*n~.able in ( x ,w). ~ e t
ti
n n d be (x,21, I) 1% drc be Loznsded, a n d let
zi?zifornzly oveY a n y finite range o f x as 1-30.
a n d let
L e t K be measurable,
$2
N. WIENER.
be finite.
Let
exist for all 1. Then
To prove this lemma, let us reflect that
By taking A large and B infinite, or --B large and -A infinite, we may
make the left-hand member uniformly small. Over any finite range (4, B)
the left-hand member tends to 0 as il-a. By combining these facts the
lemma follows. It will be noted that i t il~volvesthe existence almost
everywhere of
lmf
372
(22.22)
K G ) (z, w) d z .
W e may now return to the proof of our main theorem.
paperss, the :luthor has shown that
exists. ?1.loreover,
-56
--
Wiener (5).
<m,
111
a previous
TAUBERIAN THEOREMS.
so that
(2
&L1L:Kl(3--5)
f ( F ) d E d3;
< const.
Thus the function
W
(22.26)
fi (x) = J-W
and the analogous function
(22.27)
&(x) =
K1( 2 - 5 ) f ( ~dE
)
$ T c P ( ~ - B )f ( F ) d l
w
define functions sl (u) and sz (u), corresponding to them in the same way
in which s ( u ) corresponds to f ( x ) .
I n his previous paper,58 the author has shown that
(22.28) sin E X : .
f(x)-----sZUXdx.
X:
A+-n
Similarly
Sl (U
+
E)
-S1 (26 - E )
a00
1
A sin E X .
- - 1. i. m. JA-x ~ ~ ~ d x LKl
( -(F)- f ~( x - 5 )
7c A+w
There is no trouble in modifying (22.28) into
(22.29)
1
A-E
-- - 1.i.m.
(22.80) s (rl+r)-s(u-E)
7-c
A-+w
f
sin Ex
( T )
d5.
eium d X: .
Thus
81 ( a + & ) -s1
(1~-8)
- ( s ( ~ L + E-s(u-E))S-:K~
)
W
=
(22.31) ( 5 ) eiuf d f
A
sin E X .
1
-76[ ~ A+w
. i . r n . ~ ~ K , ( Ed)B ~ A f ( x - E ) - - x ezUxd x
A
sin E ( X - 5)
f(x-2)--------eZ""dx
X-8
In the proof of this we have used our last lemma.
ti*
84
N. WIENER.
Clearlv
Again,
sin~x
sin^(^:-8)
cos E Z U - 6 sin &to
EZw 2
cos~zu s i n ~ w
E ~ W
2 6181
max
TO i
n - x
----10
iuY
Thus and hence ly an application of (22.32)
(22.33)
t
Fs
.f
A
sin E X
-A
--I ,"'"
sin a (x - F)
x-2
dl
converges in the mean uniformly in zb as A + oo over any finite range of 5.
Hence by our lemma, we may write
s (
(22.34)
+)
1
-s ( -)
m
= -Jm&(5)
z
- ( (u
+ 6) -s (
-.
m
-8 ) )
Kl (5) eif6gd 5
c
sin~x
d ~ lA.+iW. m . S-A~ f ( ~ - 5 ) [ - - - x
-00
sin~(x-F)x-S , --]eiuxdx.
By a further use of (22.321) and our lemma, since
(22.35) 1. i. m.-
1
sinr x
z
sin s (x - 5) ehcX.lX =
x - F--I
uniformly in 5, it follows that
From (22.12) and a theorem of the previous paper,59 it follows that
jg
Wiener ('i),(5.52).
TAUBERIAN THEOREMS.
85
Combining (22.36) and (22.37), we get
Since over any finite range, by (22.16),
it results t,hat for any finite C,
am
(22.40)
1
~ s ( I L + ~ ) - s ( z-I- 1 ) 1 ~ d t ~== 0 .
€7'0
As in (22.36), me have
Vm
(22.41)
'+O
1
1f m
6 c
-W
s2 ( u
+ 8)
-s2 (tc - 6)
am
-(s(zI+E)-s(?I-E~)
This yields us
1
C '
(22.42)
liln
S+O
1J'
' --C
+
(S + a) - s
s2(u r ) - s2 (76 - el
--
(26
(11
--6))
Combining this with (22.40), we get
By our previous paper,60
and
sin % x
dr
-
" Wiener
11-('i,fif;(z+
-,.
t~)d~i'
?
(22.45)
(7), (6.62).
1
2.
LW
I2112
'"
sinqlw ,
1
a (ZL
+ E )-s2
(71
- 6) 5171.
N. WIENER.
86
Nowr
am
-
lim
€ 3 0
-300
sin% x
(x)
&J-m
IJ?
l2?b
const.
<l i m p
6
E+"
=
-
lim
E-+O
c
-j-m
00
f
-m
-A
I
f2(x+y) d ~ /
sinPs x
~~(Z-I-Y)7
~ Y , dx.
"
1 lm
roO
const.
1
1h(x)-37J1,
2
8 1
sinZE X
1
dx-zJ-mT-dx~LJ
W
f ( x - 21
A
2
singsx
1
K 2 ( @ - - 2 1 , -A K 2 ( l + y ) d y ] d F i 7 d r
f
2n + o o
(22.46)
const. 2 B
sB/ l:
&(Fj-ls
~ ( J L
-GI
F)
R
-A
23,
I /&(F)---S
boo
- const.
2
lim B , -m
--
B+m
xSw
-m
-30
where
B+w
(22.47)
=(
E+O
w
-m
A
3
A
A
2
-1
I
I
/K2(~)-&~A~~(b+y)dy/d5.
sin5x
ih(r)12--T-rlx
x
5 censt. [Jm
-CO
j
sin" x
1
1
K~(5)- 23,
sA
-A
i l z i23,
l J 2-2j i (x+y) dylfiI
2
l i , ( 5 + y ) d y I d ~ ].
As -\Ire have seen in section 1,
(22.49)
lim
i-to
,
~ , ( ~ + ~ ) jdi +y *j ) d ~
A
&J(,,
(22.45)
& ( 5 + ?1)dy1d 1
K , ( : ) - -2-1L ~-2 K 2 ( F + y ) c l Y ~( l + % ) d b ]
aw
c
Hence
lim
1
J((/K ~ ( F ) -23,L J
-
lim
-A
-,~ ( ~ + ~ ) d z l d * i ( ~ + l ~ l + l r i i )
1
2 1-
-
< const. B-+CC[ .J m
A
1
21,
1
2
K2(Ff y ) d y ] d F 1 d x
loo
/ - JA
K, (8)
--
21
-1
K2(F+y)rly d l = 0 ,
TAUBERIAN THEOREMS.
from which it immediately follows that
In combination with (22.44) and (22.45) this yields us
(22.51) lim lim
- -A-zoe+o
2~
Since
~ ~ ( u + ~ ) - s ~ ( z i - ~ )=
~ ~0.d t l
(22.52)
it results at once that
(22.53)
'
lim linl
- -[
E+O
28
C+W
~ m + ~ ~ ] i s 2 ( u + E ) - s I ( z ~ - E ) 1 2 d t=
~
0.
Combining this with (22.43), we may readily conclude that
or in view of a theorem of our previous paper, that
(22.13)
lim
T+m
1
2T
3T
r(-!~j f 2 ( x ) 1 2 c l x = 0 .
--
This establishes Theorem XXIV. I t will be noted that this theorem has
an immediate extension in the direction of Theorem XI'.
There is another Tatiberian theorem concerning strong sommability which
~ particular
we may discuss here. I t is due to Hardy and L i t t l e w ~ o d . ~The
case of it which most directly interests us is the following:
XXV. Let
THEOREM
be bozinded, and let
Hardy and Littlewood (16).
88
N. WIENER.
i t follozcs that
In the proof of this theorem, it is easy to see that we may replace the
conclusion (22.58) which we are to establish, by the equivalent
1J H
(22.59)
H
H+oo
sin zb x
Jn
o
2
---- d z j
dn = 0.
5
0
By a Tauberian theorem of the type of Theorem XI, already established
in a previous paper of the author," it follows that (22.59) is completely
eqnivalent to
sins 8 t h I 2
sin z ~ x i2
l i ~ n2
f (x) ----- d x ' d 2 6 .
(22.60)
t+O
76
Jy
F
Etb2
On the assumption that f (x) = 0 if
theorem
Jr
LL'
X
> z,we
have by the Plancherel
sin 21 5
768
-
2
76r
1% /$Jsiqrttsinax
- 1 1 a J"
zr
(22.61)
2L
sin r ( 1 cos Iix d x
JW
76 r
=i
I~~~(sinzc(x+s)-sina(~-~))dx~~~dl
-aT1_
1
-
- .
1w
L J i1+~
Jm. 2 ~
dsint~(x+~)--si~~zc(x--i))d x
2
sinaxdx
= rrE
--
sin s ti sin 2 1 x 21 x -W
--,d;51 d u
i'
I
TAUBERIAN THEOREMS.
Thus (22.59) and (2233) are equivalent t o
By the Schwarz inequality,
Thus by the boundedness of (22.55) we have uniforirlly in
ficientlg large values of N:
E
for all sof-
Tlzis can be made as small as me wish by talring N large enough.
Moreover,
N
=A
where
(22.651)
-12-1
-)
z L (l+log l x S 1 l d x ,
I
90
N. WIENER.
By (22.56), we aiay make i4 as small as we wish for a given AT by
taking E small enough. Now let N be so large that for all E ,
and then let
F
be so small that
I t ~ t ~ ithen
l l follo~vthat for this and all smaller values of
Thus (22.62) follot~s,and Theorem XSV is established.
23. Generalized types of summability. The subject-matter of the
last section leads us to interesting reflections on the notion of summability
itself. ?'he ordinary processes for summing a series are linear processes:
that is, they consist in replacing the partial sums of a series by linear
combinations of partial sums, and then investigating the ordinary limits
of these linear combinations. This much of linearity must alttrays remain
in a definition of summability, that if two summable series are added term
by term, the corresponding sums are also added, in the sense that the
new sum-series ~vill be summable to the sum of the suiiis corresponding
to the individual series.
It is possible, however, to consider summability from another standpoint,
from which linearity is not so obvious an attribute of the process. W e
may confine our attention to series, or rather to their sequences of partial
sums, which are suinmable to zero. Thus a method of summability will
sum to -4 the series whose partial sums are:
if it mill sum to zero the series tt7hose partial sums are:
\Tie may thus center our concept of summability about the notion of
slzlll-sepztence or .rlull-fiinctio?~. Whatever definition we choose for the class
of such sequences or functions, this class should be closed additively, in
the sense that the sum of any two members of the class must belong to
91
TAUBERIAN THEOREMS.
the class. It is not howwer essential that a mernber o f the class shoztld be
clzaracterized by the l~nnishingof some particular linear transform of that
member. I n this sense, we are introducing a non-linear theory of snmmability. 6 2
It is clearly a desideratum of a definition of a null-function that every
function tending to a zero limit a t infinity should be null. This requirement a t once makes the corresponding definition of summability consistent
with convergence and inclusive of it in scope. I t is not satisfied by
definitions of the type that assert that f is null if
but is satisfied by the definition that f is null if
W e have already seen that if s(u) is defined as in (22.23), this last
statement is equivalent to the assertion that
This suggests an even looser definition of a null-function, according to
which a function f ( x ) is null if
is bounded, and if
(23.07)
lim
SA
\s(21+&)-~('21-&)j2d21= 0
-
E + O 2 &
-A
for all finite A, or even if (23.07) is true for some A greater than 0 .
The first of these definitions will make f ( x ) null if (23.06) holds and
:l
(23.08)
is null, provided that
(23.09)
and
(23.10)
"
Lw+ 1
(1
rW
x
K(x-F)
f (1)dF
/ X I ( x )1'
~ ( xeiU-"
) dx # 0,
d x < c~
[-m
<rc<
m].
As far as the author knows, strong summability of various sorts is the only example
of a summability process of this sort now in the literature.
92
N. WIENER.
The second definition will have the same property, provided only (23.10)
holds for all values of t t in some neighborhood of 0 . All this follows
from the theory developed in the last section.
These two last definitions of sunlmability therefore fit in well with the
ordinary linear definitions of summability, which define the generalized
limit of a function as an expression of the form
lim
J.-w
x+"
l:
K", 5) f (5) d 5
K (x, 5) d B
on an appropriate scale of measurement. They have the great advantage
of not involving any reference to a particular kernel K (x, 5). They are,
however, restricted to functions for which (23.06) is bounded.
If we confine our attention to functions f ( x ) vanishing for negative
arguments, this difficnlty may be overcome in its turn. W e now put
throughout the half-plane 8 ( z ) > u o : Then, by analytic continuation if
)
be defined in some cases on a section of the real
necessary, ~ ( i v may
axis containing the origin. We put
giving o its boundary value along the axis of imaginaries. We now define
nnllity as in (23.06) and (23.07). If for all z with real part between
-E.
and zco s
+
rw
1 K(x) 1 @x
(23.13)
dxl'm,
we have
-d
w
(23.14) l 276,
f w e-03' - ' d x ~ ~ K ( x - - ~ ) f ( E ) d ~ = ~ ( ~ )
so that if
-03
K($) 8:d 5 is free from zeros and singularities in the left
half plane, f (x) and
SwK ( x -03
5) f ( E ) d 5 are null or not null simultane-
ously. Wit,h an appropriate definition of "analytic continuation", zeros
and singularities not a t the origin are of no concern, provided the function
TAUBERIAN THEOREMS.
93
J-1
K(5) ez' d 5 is representible on a single-sheeted Riemann surface in
the left half-plane.
It will be noted that all tlie definitions liere given of the nullity of f(x)
depend on the scale cl~osenfor x. If for all large x ,
y'(2) < const. for x < A
Y (A)
and
then it a t once follows that
A similar study of the effect of a change of scale 011 nullity of other
types would be of interest.
24. Some unsolved problems. In a piece of work of the ariibitious
length of the present, it is perhaps worth while to point out to the reader
promising future directions of research. The follo~vi~lg
remarks may therefore not be amiss.
(1) The closure of the set of translations of a given function 11as been
investigated in class LI and in class L2. The methods of proof have been
widely different in the two cases, but both results may be stated in a single
formulation, that the set of translations of f(x) is closed in the appropriate
class when and only when the Fourier transform of f(x), when properly
defined and chosen, has no zeros. This formulation continues to constitute
a reasonable proposition for L.,, where p is intermediate between 1 end 2,
or even exterior to this interval. Is this proposition true? Certainly, for
neither of the special cases already given is the method of proof extensible
without serious modification. My own suspicion is that the general tlieorem
is a t least true for 1 5 p 5 2 .
(2) Obviously the power of Tauberian theorems in number theory has
not been exhausted. I s there any Tauberian tlieorem which will reach
froni one complex ordinate to another, and enable us to handle the more
refined forms of the prime number theorem?
(3) 111 particular, can we make a direct study of the closure of the set
of all polynomials in the functions
94
-N. WIENER.
and thus attack directly the problem of the zeros of the zeta function in
the critical strip?
(4) I n section 22, the conditions to which K, (z) and K, (x) are subjected
in Theorem XXIV are probably needlessly stringent. What is the best
possible theorem in this connection?
( 5 ) The "quasi-Tauberian" theorem of sectioi~21 has not yet been exploited to the full. In particular, in the discussion of the relations between
Riesz summability, Cesiro surnmability, Holder summability, and the like,
it is extremely desirable to have theorems of this type in which the kernels
are not of the form K ( x - y), but are in some sense near.2~of this form.
Here the cruder theorems, depending solely on the use of dominant functions, are probably not difficult to elicit. On the other hand, the more
refined ones will almost certainly require the full armament of Carleman's
theory of singular integral equations and of the modern von NeumannStone63 calculus of operators. Indeed, the time will soon come when the
entire Tauberian theory must be reconsidered from the point of view of
this calculus.
BIBLIOGRAPHY.
The present list of memoirs, while i t cannot in the nature of things be complete,
represents an attempt to bring together in one place as much as possible of the literature
of Tauberian theorems. Up to 1915, we have been able to draw on the bibliography of
the Cambridge tract by Hardy and 11. Riesz on The General Theory of Dirichlet Series,
and up to about 1924, on Smail's History aiid Synopsis of the Theory of Summable
Infinite Processes. The bibliography also contains certain memoirs mentioned in the notes
which are not of themselves Tauberian.
K. Ananda-Rau.
(1) d note on a theorem of Mr. Hardy's. Proc. Lond. Math. Soc., (2) 1 7 (1918), 334-336.
(2) On Lambert's series. Proc. Lond. Math. Soc., (2) 1 9 (1921). 1-20.
(3) On the relation between the convergence of a series and its sunlmability by Ceshro's
means. Jour. Indian Math. Soc., 1 5 (1923-5), 264-268.
(4) On Dirichlet's series with positive coefficients. Rendiconti di Palermo, 54 (1930),
488-462.
S. Bochner and G. H. Hardy.
(1) Note on two theorems of Norbert Wiener. Jour. Lond. Math. Soc., 1 (1926), 240-242.
E. Bortolotti.
(1) Sulle condizioni di applicabilith e di coerenza dei processi di sommazione asiiltotica
di algoritmi infiniti. Rend. del R. d c . di Bologna, 31 (1926-7), 33-41.
G 3 Von
Neumann (1); Stone (1).
TAUBERI AN THEOREMS.
95
S. Bosanquet.
(1) The sumlnability of Fourier series. Math. Gaz., 1 6 (1931), 293-296.
(2) On the summability of Fourier series. Proc. Lond. Math. Soc., (2) 31 (1930), 144-164.
L. S. Bosanquet aiid E. H. Linfoot.
(1) On the zero order summability of Fourier series. Jour. Lond. Math. Soc., 6 (1931),
117-126.
(2) Generalized means and the summability of Fourier series. Quarterly Journal (Oxford
series), 2 (1931), 207-229.
H. E. Bray.
(1)Elementary properties of the Stieltjes integral. Ann. of Ilath., 20 (1919), 176-186.
T. J. I'a. Bromnlich.
(1) An introduction to the theory of infinite series. London 1908. Cf. especially p. 251
(2) On the limits of certain infinite series aiid integrals. Math. Ann. 65, (1908), 350-369.
Anna Caldarera.
(1) Su talune esteilzioni dei criteri di convergeiiza di ITarrlg e Landau. S o t e e memorie
di mat. (Catania), 2 (1923), 77-98.
T. Carlemall.
(1) A theorem concerning Fourier series. Proc. Loild. Math. Poc.. (2) 21 (1923), 483-492.
11. S. C'arslaw.
(1) Introduction to tlie theory of Fourier series and integrals. New edition London 1921.
Cf. especially pp. 150-154, 434-243.
S. Chapman.
(1) On non-integral orders of summability of series and integrals. Proc. Lond. Math.
Soc., (2) 9 (1910-11), 369-409. Cf. 11. 374, second note.
11. Cipolla.
(1) Criteri di convergenza ridncibili a rluello di Hardy-Landau. Naples Rendiconti (3a),
27 (1921), 28-37.
P. J. Daniell.
(1) A general form of integral. Ann. of Math., 19 (1917), 279-294.
P. Dienes.
(1) Snr la sommabilitB de la sBrie de Taylor. C. R., 153 (1911), 802-805.
G. Doetsch.
(1) Ein Konvergenzkriterium fur Integrale. Math. Ann., 82 (1920), 68-82.
(2) Die Integrodiffereiitialgleichungen vom Faltungstypus. Math. Ann., 89 (1923), 192-207.
(3) Satze von Tauberschem Charakter im Gebiet der Laplace- und Stieltjes Transformation.
Berliner Berichte, Phys.-Math. Klasse (1930), 144-157.
(4) Dber den Zusammenhang zwischen Abelscher und Borelscher Summabilitat. Math.
Ann., 104 (1931), 403-414.
P . Fatou.
(1) SBries trigonometriques et series de Taylor. Acta Math., 30 (1906), 335-400. (ThBse,
Paris 1907.)
L. FejBr.
(1) Pourierreihe und Potenzreihe. lfonatshefte f. JIath., 28 (1917), 64-76.
96
(1) Visegklatok a Fourier sorokrol.
N. WIENER.
$1. Fekete.
Math. 6s Termesz. ~ r t . ,34 (1916).
M. Fujiwara.
(1) Uber die Verallgemeinerung des Tauberschen Satees auf Doppelreihen. Science reports
of the Tbhoku national uliiversity, Sendai, Japan, 8 (1919), 43-50.
(2) Uber suiumierbare Reihen und Iategrale. Tbhoku math. jour., 15 (1919), 323-3-29.
(3) Ein Sate iiber die Borelsche Sumn~ation. Tbhoku niath. jour., 17 (1920), 339-343.
J. Hadamard.
(1) Sur la distribution des zeros de la fonction f(s)
Bull. Soc. Math. France, 24 (1896), 199-220.
et ses conseque~~ces
aritlimetiques.
(3. H. Hardy.
(1) Researches in the theory of divergent series and divergent integrals. Quarterly journal,
38 (1904), 22-66.
(2) Theorems relating to the summability and convergence of slowly oscillating series.
Proc. Lond. Math. Soc., (2) 8 (1909), 301-320.
(3) On the multiplication of Dirichlet series.
Proc. Lond. Math. Soc., (2) 10 (1912),
396-40.5.
(4) On the summability of Fourier's series. Proc. Lond. Math. Soc., (2) 12 (1913), 365-372.
(5) An extension of a theorem on oscillating series. Proc. Lond. Nath. Soc., (2), 1 2 (1913),
174-180.
(6) Note on Lambert's series. Proc. Lond. l a t h . Soc., (2) 1 3 (1913), 192-198.
(7) The application of Abel's niethod of summation to Dirichlet series. Quarterly journal,
47 (1916), 176-192.
(8) The second theorem of consistency for summable series. Proc. Lond. Math. Soc., (2)
15 (1916), 72-88.
(9) On certain criteria for the convergence of the Fourier series of a continuous function.
Messenger of mathematics, 49 (1920), 149-155.
(10) The sumnlability of a Fourier series by logarithmic means. Quarterly Journal (Oxford
series), 2 (1931), 107-113.
G. H. Hardy and J. E. Littlewood.
(1) Contributions to the arithmetic theory of series. Proc. Lond. Math. Soc., (2) 11 (1913)'
411-478.
(2) The relations between Borel's and Cesiro's methods of summation. Proc. Lond. Math.
SOC.,(2) 11 (1913), 1-16.
(3) Sur la serie de Fourier d'une fonction a carre sommable. C. R. April 28, 1913.
(4) Tauberian theorems concerning power series and Dirichlet's series whose coefficients
are positive. Proc. Lond. Math. Soc., (2) 1 3 (1913), 174-191.
(5) Some theorems concerning Dirichlet's series. Mess. math., 43 (1914), 134-147.
(6) Theorems concerning the summability of series by Borel's exponential method.
Rendiconti di Palermo, 41 (1916), 36-53.
(7) Sur la convergence des series de Fourier et des series de Taylor. C. R. Dec. 24, 1917.
(8) The Riemann zeta-function and the theory of the distribution of primes. Acta Math.,
4 1 (1918), 119-196.
(9) On the Fourier series of a bounded function. Proc. Lond. Math. Soc., (2) 17 (1918),
xiii-xv. (Abstract).
(10) Abel's theorem and its converse. Proc. Lond. Math. Soc., (2) 1 8 (1920), 205-236.
97
TAUBERIAN THEOREMS. (11) On a Tauberian theorem for Lambert's series, and some fundamental theorems in the
analytic theory of numbers. Proc. Lond. Math. Soc., (2) 19 (1921), 21-29.
(12) Solution of the Cesaro summability problem for power-series and Fourier series.
Math. Ztschr., 1 9 (1923), 67-96.
(13) Abel's theorem and its converse (II). Proc. Lond. Math. Soc., (2) 22 (1924), 254-269.
(14) The allied series of a Fourier series. Proc. Lond. Math. Soc., 24 (1925), 211-246.
(15) The strong summability of Fourier series. Proc. Lond. Math. Soc., (2) 26 (1926), 273-286.
(16) Elementary theorems concerning power series with positive coefficients and momentconstants of positive functions. Crelle, 157 (1926), 141-158.
(17) A further note on the converse of Abel's theorem. I'roc. Lond. Math. Soc., (2) 25 (1926),
219-236.
(18) Notes on the theory of series (VII): on Young's convergence criterion for Fourier series.
Proc. Lond. Math. Soc., (2) 28 (1928) 301-311.
(19) Notes on the theory of series (XI): on Tauberian theorems. Proc. Lond. Math. Soc.,
(2) 30 (1929), 23-37.
(20) Notes on the theory of series (111): on the summability of the Fourier series of a nearly
continuous function. Proc. Camb. Phil. Soc., 23 (1527), 681-684.
(21) The equivalence of certain integral means. Proc. Lond. Math. Soc. records, (2) 22 (1924),
xl-xliii.
G. H. Hardy and M. Riesz.
(1) The general theory of Dirichlet series. Cambridge Tract in Mathematics and Rlathematical Physics, No. 18. Cambridge 1915.
G. H. Hardy and E. C. Titchmarsh.
(1) Self-reciprocal functions. Quarterly journal (Oxford series), 1 (1930), 196-232.
(1) A trivial Tauberian theorem.
W. A. Hurwitz.
Bull. Am. Math. Soc., 32 (1926), 77-82.
8. Ikehara.
(1) An extension of Landau's theorem in the analytic theory of numbers.
llath, and Phys. of the Mass. Inst. of Technology, 1 0 (1931), 1-12.
Journal of
S. Izumi.
(1) A generalization of Tauber's theorem. Proc. Imperial Acad. Japan, 5 (1929), 57-59.
M. Jacob.
(1) ijber den Eindeutigkeitssatz in der Theorie der verallgemeinerten trigonometrischen
Integrale. Math. Ann., 100 (1926-7), 278-294.
(2) Uber ein Theorem von Bochner-Hardy-Wiener. Jour. Lond. Math. Soc., 3 (1928), 182-187.
J. Karamata.
(1) Sur le mode de croissance regulikre des fonctions. RIathematica (Cluj, Roumania),
4 (1930), 38-53.
(2) Ober die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes. Math.
Zeitschr., 32 (1930), 319-320.
(3) Neuer Beweis und Verallgeineinerung einiger Tauberian-Siltze. Math. Zeitschr., 33
(1931), 294-300.
(4) Neuer Beweis und Verallgemeinerung der Tauberschen Satze, welche die Laplacesche
und Stieltjessche Transformation betreffen. Crelle, 164 (1931), 2 7 4 0 .
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N. WIENER.
A. Kienast.
(1) Extension to other series of Abel's and Tauber's theorems on power series. Proc.
Lond. Math. Soc., (2) 25 (1926), 45-52.
K. Kiiopp.
(1) Neuere Untersuchungen in der Theorie der divergenten Reihen. Jahresber. d. deutschen
Mathematikervereinigung, 32 (19231, 43-67.
(2) Theorie uud Anwendung der unendlichen Reihen. Berlin, Springer, 1922. Cl~np.SIII.
E. Landau.
(1) Uber die Konvergenz einiger Klassen von unendlichen Reihen am Rande des Konvergenzgebietes. llonatsh. f. Math. und Phys.. 18 (1907), 8-28.
(2) Handbuch der Verteilung der Primzahlen. Leipzig 1909, 2v.
(3) Uber die Bedeutung einiger neuerer Grenzwertsatze der Hrrreil Hardy und Axer.
Prace mat.-Fiz., 21 (1910), 97-177.
(4) Uber einen Satz des Ilerrn Littlewood. Rendiconti di Palermo, 35 (1913), 265-276.
(5) Ein neues Konvergenzkriterium fur Integrale. Munch. Sitzber., 43 (1913), 461-467.
(6) Darstellung und Begrundung einiger neuerer Ergebnisse der Funktionentheorie. Berlin
1916, 2nd ed., 1930.
(7) Sobre 10s n6ineros priinos en progressi6n aritmetica. Rev. Xat. Hispano-Americana,
4 (1923), 1-16, 33-44.
(8) Vorlesungeil uber Zahlentheorie. Leipzig 1927, 3v.
H. Lebesgue.
(1) Recherche3 Tur la convergence dcs series de Fourier.
JIath. Ann.. 61 (1905), 251-280.
P. LBvy.
(1) Sur les conditions d'application et sur la regularit6 des proc6d6s de sommation des
series divergentes. Hull. Soc. Math. de France, 54 (1926), 1-25.
S. B. Littauer.
(1) On n theorem of Jacob. Jour. Lond. Math. Soc., 4 (1929), 226-231.
(2) A new Tauberian theorem with applicatioil to the summability of Fourier series and
integrals. Jour. Nath. Phys. Nass. Insti. Tech.. 6 (1928), 216-234.
J. E. Littlewood.
(1) On the converse of Abel's theorem on power series. Proc. Lond. Math. Soc., (2) 9
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H. J. Nellin.
(1) Die Theorie der asymptotischen Reihen vom Standpunkte der reziproken Funktionen
und Integrale. Ann. Ac. Sci. Fennicae, (A) 1 8 (1922), No. 4, 108 p.
G. Jlignosi.
(1) Inversione d'un teorema sul rapport0 delle medie (Cp) di due serie. Kaples Rendiconti
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(1) Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren.
Math. Ann., lO'2
(1930), 49-131.
R. E. A. C. Paley.
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99
M. Plancherel.
(1) Contribution X l'etude de la representation d'une fonctioii arbitraire par des iilt6grxles
dkfiniez. Rendiconti di Palermo. 30 (19101, 289-335.
A. Priiigsheiiri.
(I) Uber das Verhalten von Potenzreihen aui dem Koiirergenzkreise. Munch. Ber.. 30
(1900), 43-100.
(2) Ober die Divergeilz gewisser Potenzreihen an der Konvergenzgrenze. M. Ber., 3 1
(1901), 505-524.
(3) Uber eine Konvergenzbedingung fur unendliche Reihen, die durch iterierte Mittelbildung reduzibel sind. 11. Ber., 50 (1920). 275-284.
M. Riesz.
(1) Ulie methode de sornmatioii 6quivaleiite & la lnethode des moyen~lesarithmetiques.
C. R., 12 June 1911.
(.L) Uber einen Satz des Herrn Fatou. Crelle, 140 (1911) 89-99.
R. Schmidt.
(1) Uber das Borelsche Summierungsverfahren. Schriften der Konigsberger gelehrten
Gesellschaft, 1 (1925). 202-256.
(2) Uber divergente Folgen und lineare Mittelbildungen. M. Ztsch., 22 (1925), 89-152.
(1) Uber Dirichletsche Reiheii.
W. Schnee.
Rend. di Palermo, 27 (1909), 87-116.
J. Schur.
(1) Ober lineare Transforlnationen in der Theorie der unendlichen Reihen. ('relle, 151
(1921), 79-111.
L. L. Smail.
(1) History and synopsis of the theory of summable infinite processes. V. of Oregon
yubl.. 2 (Feb. 1925), So. 8.
M. H. Stone.
(I) Linear transformations in Hilbert space 111. Operational methods and group theory.
Proc. Mat. .kcad. Sci., 1 G (1930). 172-175.
(1) 011 a theorem of Carleinsn.
0. G. Sutton.
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