PATTERN INTEGRATION WITH IMPROPER
RIEMANN INTEGRALS
R. E. CARR
1. Introduction.
In a previous paper [l]1 we have recently introduced and discussed a type of integration, which we have called
pattern integration, and have established the following theorem:
Theorem
1.1. Let f(x) be (proper) Riemann integrable, 0 = x = \.
Let (k-l)/ne^ek/n
(k = l, 2, 3, • • • , n). Let the pattern P be
characterized by the dyadic number
t = 0.aia2a3
• • • an • - • (2),
such that
1
n
lim — /. ak = a.
n->» n fc_i
Then the pattern integral
(P) f f(x)dx m lim - ¿ a*/(íln))= a(R) f f(x)dx.
J o
»-*"
n ic**i
J o
It seems desirable to extend this result, if possible, to improper
Riemann integrals. In this paper we shall confine our remarks to
improper integrals of the form
J
<f f(x)dxm Um(R) f f(x)dx.
0
€-*>+
(R) in front of any integral will indicate
J e
a proper Riemann
integral.
2. The improper integral defined as the limit of a sum. We first
consider the case where ak = \ (k = i, 2, 3, • • • ). A fundamental
difficulty not encountered
in the previous paper is apparent from
the definition of ¿f\f(x)dx. Bromwich and Hardy have considered
this problem [2]. As they point out, the ordinary definition of this
improper integral is by means of a double (repeated) limit; and in
order to replace this limit by the single limit,
lim -±f(&\
n-»°°
n k=i
Received by the editors December 11, 1950.
1 Numbers in brackets refer to the bibliography at the end of the paper.
925
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926
R. E. CARR
[December
some restrictions are certainly necessary. In the first place, if 0 <£Ín>
^1/w, we can clearly choose £Ín) so that | (l/«)/(£íB))|
is greater
than any assigned number. It is then suggested that the most obvious
choice is ^ —k/n, and with this restriction they establish:
Theorem 2.1. Let f(x) be positive and tend steadily to <x>as x tends
to zero. Let J"¡j"(x)dx be convergent. Then
lim - Z f (-)=(£
n-»«> n t=i
\ n /
Jo
J(x)dx.
The proof rests on the fact that for all n,
(2.1)
1
n
-Z/(-)
» M
while the difference
/ k\
\»/
f1
1 n_1 / k \
= (P-) /(*)<**
= -Z/(-),
J 1/n
between
(2-2)
» fc-l
\ » /
the two sums in (2.1) is
tMt)-*»}-
That the limit of (2.2), as »—»<», is zero, follows from the well
known result that if f(x) steadily increases as x decreases and
flf(x)dx
is convergent,
then \\mx^o+xf(x) =0. From this the conclusion is immediate.
It appears that the following extension of the above theorem is
new, although the theorem is quite elementary.
Theorem 2.2. Let f(x) be positive and tend steadily to « as x tends
to zero. Let flf(x)dx
be convergent. Let (l/Mn) g£r°á(l/»)
(Af=T
and fixed), and (k-i)/n^)g(k/n)
lim -¿/(£ÍB))
n-">
n fc_i
(k = 2, 3, • ■ • , n). Then
=<£ f(x)dx.
J o
Proof. Define
A..±ÈJ(±)
n ¡t=i \n /
and
n I \Mn/
*„2 V n /'
n I \MnJ
t=i \ n /
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)
I95i]
PATTERN INTEGRATION
927
Then
An = -tf(Ún))^Bn.
n k=i
But,
clearly,
lim„,„
An = limn^x Bn = ¿flf(x)dx,
and
our conclusion
follows.
A sharper
criterion,
(2.3)
in order that the relation
lim — ¿/(—)
n->» n k=i
\n
= (f f(x)dx
/
Jo
should hold whenever the improper integral on the right is convergent,
has been given by Wintner [3] in the following result:
Theorem 2.3. Iff(x) is of bounded variation on every interval e = x = 1,
where e>0, and behaves, as x—»0, so as to satisfy the restriction
j
| df(x) | = o(e~1),
then the convergence of ¿P\f(x)dx
(2.4)
implies that
limt£/i>)
«-»0+
teál
exists and equals ^\f(x)dx.
The result (2.3) would follow from (2.4) by letting e = l/n.
The following
extension
of Wintner's
result
appears
to be new:
Theorem 2.4. Let f(x) be of bounded variation on every interval
e = x=l, where e>0, and let f(x) behave, as x—»0+, so as to satisfy the
restriction
J
(2.5)
Let%i= \/Mn (M=\
| df(x)| = «(e-1).
and fixed), and (k-i)/n^^
■• • , n). Let ¿flf(x)dx exist.
Then
lim -¿/(|£n>)
n->»
Proof.
(2.6)
n fc=,i
From the hypothesis
=<£
f(x)d*
J o
it follows that
lim xf(x) = 0.
l->0
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^(k/n)
(k = 2, 3,
928
R. E. CARR
[December
For,
«1/(1)
-/M| -« f'á/wláÉ V\df(x)
From Wintner's
result, we know
lim — ¿/(—)
n-»»
«
4=1
\ » /
=^
./
/(*)<**■
0
Consider
ft/(£r)-i£/(i.)}.
I » *-i
w jb.1 \ n I)
For all n,
±f(án))-±f(-)\é±\ñán))-f(-)\
k=i
k=i
\n/
\
<
k-i\
\n/
\
f ld/(*)l
+|/(xr)l
+k(~)lI \Mn/ I
\ n/ I
J d/n)
Hence, using (2.5) and (2.6),
lim
i¿/(i:-')-i¿/W|
n k=i
n k~i \n/
\
=o,
and our conclusion follows.
It should be noted that a number of theorems are available in the
"converse" direction, that is, in which the convergence of the improper integral ¿flf(x)dx follows from the knowledge of the existence
of Iimí_o+e2»«ái /(Me)- Ln this paper, however, we are not concerned with such results.
3. A pattern integral theorem. To hope for a complete extension
of Theorem 1.1 to improper integrals of the form ß\f(x)dx
is too
much. Counter examples are not difficult to construct. For example,
see §5 of the paper by Bromwich and Hardy [2].
The following theorem partially extends the earlier results.
Theorem 3.1. Let f(x) be positive and tend steadily to » as x tends
to zero. Let f\f(x)dx
be convergent. Let (l/Mn) álrá
(l/n) (M^l
and fixed), and (k —l)/ngi£)
gk/n
(k = 2, 3, - • ■, n). Let the pattern
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1951]
PATTERNINTEGRATION
929
P be characterized by the dyadic number
t = 0.aia2u!3
• • • an • ■ ■ (2),
such that
1 Ï
lim
— / '. cu = a.
n-.»
n k=i
Then
(P) £ f(x)dx »lim
J
Proof.
0
n-">
— ¿ akf(tT) = a<£ f(x)dx.
n k=l
J
Define fN(x)=f(\/MN),
—x = l. For all values of N>0,
(3.1)
0
0 = x<l/iV, fN(x)=f(x),
(P)/¿/at(x)¿x
Í/N
exists and
(P) f fN(x)dx= a(R) f fN(x)dx.
Jo
Consider
Jo
the expression
(3.2)
-¿«¿/(a"')n *,=!
Let the subinterval
lim«..«, \(n)/n
= \/N.)
in which Í/N
Hence
occurs be the X(w)th. (Then
(3.2) can be written
as \n+\\n
— III„
+ IV„, where
I» = —«x(n)/($\(7.)),
n
1 x(n)
II» = — ¿2 otkftr(h ),
n k=i
1 M»)-l
( i
. .
in. = —
Z «*/*(&), iv»= —
Z «*/(&).
n k=i
n fo^i
Let us consider
the behavior
Clearly /(fíjij) ^f(l/Mn),
From (3.1), we see that
of I„, II„, III„,
lim II, = a(R) i' fs(x)dx = a i±.f(-L-)
»-.«
and IV„, as n—>*>.
and since limI.,o+ x/(x)=0,
Jo
\N
\MN/
lim„,„ I„ = 0.
+ (R) f f(x)dx\ .
J un
)
Now
/
1 \X(») - 1
\MN /
n
1
X(«) -
*<£-»
1 kmi
Hence,
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1
«
(n)
930
R. E. CARR
[December
limIIIn = a—/(-J.
N
\MN )
We have
1 X(n)-1
. ,
iv. s -n *_i
E /(«*)•
Hence
ruv
1
0 = IVng*
Jo
for all n. We then see that,
oscillates between
(3.3)
/ 1 \
f(x)dx + -f(~)
n
\Mn/
as w—><»,the expression
(3.2) at worst
a(R) f /(*)<**
J l/N
and
/.
1
/(*)<fx+ (b/«1/iV f(x)dx.
UN
J
0
But this is true for all Ar>0, so letting N—>k>, we see from (3.3) and
(3.4) that
Um — ¿«»/({î°)
= a(Ä)^"
/(x)dx,
which is the desired result.
4. Miscellaneous
results. The following is an extension of Theorem
3.1.
Theorem 4.1. Let ¿f\f(x)dx be convergent. Let there be a g(x) with
the following properties :
1. In the interval 0<x^r,
where 0<rgl,
g(x) is positive and tends
steadily to w as x tends to zero.
2. $og(x)dx is convergent.
Let |/(x)|^g(x),
0<x = r. Let l/Mn^^^l/n
(Af=l and fixed),
and (i-1)/»^"1^/»
(k = 2, 3, ■ ■ ■, n). Let the pattern P be characterized by the dyadic number
t = O.aia2o!3
• ■ ■ an ■ ■ ■ (2),
such that
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1951]
PATTERN INTEGRATION
931
1 "
Hm — ¿I ah = a.
n-»»
n k=l
Then,
(P) (h f(x)dx m lim — Z oLkf(kk
) = a(b f(x)dx.
J 0
»-»» n k-i
J 0
Since the proof is similar to that of Theorem
It is of interest to note that if one chooses
1
3.1, it will be omitted.
1
f(x) = — COS— ;
X
then ¿flf(x)dx
converges,
X
but
(4.1)
lim -tf(án))
n->"
n h~i
fails to exist.
Define
1 / 1\
1 " ¡kAn = — /( — )+ — E /(-J
n
\Mn/
n k-i
\
1\
n
(M ^ 1 and fixed),
/
*--!■£/(A
n k^i
\ n /
Then An-Bn = (l/n){f(l/Mn)-f(l)}.
lim M -
n-<»
which is certainly
Mn
If (4.1) exists,
/(-)
\Mn/
not the case when/(x)
= 0,
= (1/x) cos (1/x).
Bibliography
1. R. E. Carr and J. D. Hill, Pattern
integration,
Proceedings
of the American
Mathematical Society vol. 2 (1951) pp. 242-245.
2. T. J. I'a Bromwich and G. H. Hardy, The definition of an infinite integral as
the limit of a finite or infinite series, The Quarterly Journal of Pure and Applied Mathe-
matics vol. 39 (1907-1908) pp. 222-240.
3. A. Wintner,
The sum formula of Euler-Maclaurin
and the inversions of Fourier
and Möbius,Amer. J. Math. vol. 69 (1947)pp. 685-708.
Michigan State College
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