PROCEEDINGS
OF THE
AMERICAN
MATHEMATICAL
SOCIETY
Volume 95, Number 2, October 1985
COMPACTNESS
IN
L 2 AND
THE FOURIER TRANSFORM
ROBERT L. PEGO
The Riesz- Tamarkin compactness theorem in LP (Rn) employs notions of
LP-equicontinuity and uniform LP-decay at 00. When 1 ~ p ~ 2, we show that these
notions correspond under the Fourier transform, and establish new necessary and
sufficient criteria for compactness in L2(Rn).
ABSTRACT.
An oft-quoted classical result characterizing
M. Riesz and J. D. Tamarkin (see [1, 2, 4]):
A bounded subset K of LP(Rn),
THEOREM.
and only if
(1) JR" If(x
(II)
J1xl>R
+ y)
If(x)lP
- f(x)IP
compact
1< p <
sets in LP(Rn) is due to
00, is conditionally compact if
dx -7 0 as y -7 0 uniformly for fin K, and
dx -7 0 as R -7 00 uniformly for fin K.
Property (1) is a uniform smoothness property. By analogy with the terminology of
Arzela-Ascoli, we say the functions in K are LP-equicontinuous if (I) holds. Property
(II) is a uniform decay property. The connection between smoothness and decay
through the Fourier transform has been well explored [6]. Yet the following nice
equivalence seems to be new:
THEOREM
1. Let K be a bounded subset of L 2(Rn) and let K be the Fourier transform
of K, K = {]II E K}. The functions of K are L 2-equicontinuous if and only if the
decay uniformly in L 2, and vice versa. That is, K satisfies (I) in L 2 if and
functions of
only if K satisfies (II) in L2, and vice versa.
k
Combining this result with the Riesz- Tamarkin theorem, we obtain two alternative
characterizations
of compact sets in L2(Rn):
2. A bounded subset K of L 2 (Rn) is conditionally compact if and only if
J If(x + y) - f(x)12 dx -7 0 as y -7 0, and J Il(g + w) - lcg)12 dg -7 0 as w -7 0,
both uniformly for fin K.
THEOREM
THEOREM
J1XI>R
If(x)/2
3. A bounded subset K of L2(Rn)
dx -7 0 and
J11;1>R
is conditionally compact if and only if
Ilcg)12 dg -7 0 as R -7 00, both uniformly for fin K.
Received by the editors November 7,1984.
1980 Mathematics Subject Classification. Primary 46E30, 42B99.
Key words and phrases. Compactness, Fourier transform, LP, L 2, LP-equicontinuity.
©1985 American Mathematical Society
0002-9939/85 $1.00 + $.25 per page
252
COMPACTNESS IN L2
253
Theorem 1 is an easy consequence of the theorem below, which offers some results
in L P, 1 ~ p ~ 2.
THEOREM4. Let K be a bounded subset of LP, 1 ~ p ~ 2. If K satisfies (I) (resp.
(II))
in LP, then K satisfies (II) (resp. (I)) in Lq, where l/p + l/q = 1. (If q = 00,
conditions (I) and (II) are to be stated in the obvious way using the sup norm.)
Let us set our notation and recall some basic results. For f E L1(Rn),
Recall [3]:
(1) The Fourier transform above extends to a bounded linear map f ~ J from LP
to Lq, for 1 ~ p ~ 2 and l/p + l/q = 1, so IIJllq ~ Cpllfllp forfin LP.
(2) Forfin
LP, win Rn, we have [e-iWXf(x)(a)
= Ja + w) in Lq.
(3) For fin LP, t/; in the Schwartz classY, (f* t/;)Aa) = Ja)~a)
in Lq, where
f
* t/;(x) = !R" f(x
- y)t/;(y)
dy.
PROOFOF THEOREM4. First, we assume K satisfies (II) in LP. Let M be a bound
for KinLP.
ForfinK,
fU
+
w) - f(~)
=
[(e-iWoX - l)f(x)r(~),
whence
Let e > O. Because of (II) we may choose R so large that the second term here is less
than !(e/CpF
independent of finK.
Then since !lxl~R(lxllf(x)IF
dx ~ (RMF
for f in K, we have IIfa + w) - J(~)lIq < E if w is sufficiently small, IwlP
< J.(e/CpRMF,
independent offin K. So k satisfies (I) in Lq.
Now assume K satisfies (I) in LP. We seek to show that functions in .k decay
so that t/;R and
uniformly in Lq. Let t/;(x) = (21T)-n/2e-1xI2/2, t/;R(X) = t/;(Rx)Rn,
~Ra)
= ~a/R)
are in Y, with ~a) = e-I~12/2, ~R(O) = !t/;R(y) dy = 1. Now for
1
A
I~I~ 2R, "2 ~ 1 - t/;R(O, so for f E K,
21(
f
1~1>2R
/J(O/q
d~
) l/q ~ IIJ(O(l - ~R(O)lIq
~ Cpllf(x)
- f*
~ cp[f If (j(x)
t/;R(x)/Ip
-f(x
By Jensen's inequality and Fubini's theorem, this is
- Y))'h(Y)
dylP
dx
r
254
R. L. PEGO
Now define a uniform LP modulus of continuity for K,
H(y)
By (I), H(y)
= fEK
supf
lJ(x)-j(x--y)(dx.
~ 0 asy ~ 0, and H(y)
[ ~~1>2R
IJ(Olq dg ]l/q ~
~ (2M)P
2Cp [
f
for ally. From above, we have
H(~).p(y)
dy ]l/P ~ 0
as R ~ 00 uniformly for j in K. Hence, K satisfies (II).
We conclude with a small application, which illustrates a principle known in
information theory (see [5]) that an operator in L2 that is "band limited and time
limited" is compact.
bounded functions on Rn which satisfy lim1xl->oocpi(X)
= 0,
denote the multiplication operator on L 2 given by u( x) ~
i = 1,2. Let F denote the Fourier transform operator u ~ Fu = u.
cpi(X )u(x),
Define an operator Ton L2 by T = cp1Fcp2' Assume cpl(X) is continuous.
Fix any
i
CPl(X),
= 1,2, and let
CP2(X)
cpi
PROPOSITION.T is a compact operator on L2•
PROOF.Let K be a bounded set in L 2. Clearly, the set cp2K has the uniform decay
property (II) in L2. From Theorem 1, the set FCP2K is L2-equicontinuous (has
property (I». The set TK = cpl FCP2 is also L 2-equicontinuous, and also has the
uniform decay property (II). By Riesz-Tamarkin, it follows that TK is precompact.
Q.E.D.
ACKNOWLEDGEMENT.
The author thanks Jonathan Goodman for pointing out this
application.
REFERENCES
1. R. Adams, Sobolev spaces, Academic Press, New York, 1975.
2. N. Dunford and J. T. Schwartz, Linear operators, Part I, Wiley-Interscience, New York, 1966.
3. E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press,
Princeton N. J., 1971.
4. K. Yosida, Functional analysis, 2nd ed., Springer-Verlag, New York, 1968.
5. H. J. Landau and H. O. Pollack, Prolate spherodial wave functions, Fourier analysis, and uncertainty.
III, Bell System Tech. J. 41 (1962), 1295-1336.
6. M. Murata, A theorem of Liouville type for partial differential equations with constant coefficients, J.
Fac. Sci. Univ. Tokyo IA 21 (1974), 395-404.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN, ANN ARBOR, MICHIGAN 48109