Methods of modern mathematical physics volume 2 fourier analysis, self adjointness michael reed, barry simon

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METHODS OF
MODERN M ATHEM ATICAL PH YSICS
FOURIER ANALYSIS. SELF-ADJOINTNESS

M ICHAEL REED

BARRY SIMON

D epartm ent o f M a t hem atíes
D uke U niversity

D epartm ents o f M athem atics
and P hysics
Princeton U niversity

ACADEMIC P R ESS , INC.
Harcourt Braca Jovanovich, Publishers
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R eed, M ichael.
M ethods o f m o d ern m ath e m a tic a l physics.
In clu d es b ib lio g rap h ical references.
C O N T E N T S: v. 1. F u n c tio n a l a n a ly s is .-v . 2. F ourier analysis, s e lf-a d jo in tn e ss .I.
M athem atical p h ysics.
I.

S im ó n , B a rr y .jo in t
a u th o r.
II.
T itle.
Q C 20 .R 3 7 1972
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AMS (M O S) 1 9 7 0 S u b je c t C lassifications: 4 2 -0 2 , 4 7 -0 2

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U n ite d K in g d o m E d itio n p u b lis h e d b y
A C A D E M IC P R E S S , I N C . ( L O N D O N ) L T D .


T o our parents
H elen and Gerald R ee d

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M innie and H y Simón

Preface

M ire R eed
B arry S imón

June 1975

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This volume continúes our series of texts devoted to functional analysis
methods in m athem atical physics. In Volume I we announced a table of
contents for Volume II. However, in the preparation of the m aterial it
became clear th at we would be unable to treat the subject m atter in
sufficient depth in one volume. Thus, the volume contains Chapters IX and X ;
we expect that a third volume will appear in the near future containing
the rest of the m aterial announced as “ Analysis of O perators.” We hope
to continué this series with an additional volume on algebraic methods.
It gives us pleasure to thank many individuáis:
E. Nelson for a critical reading of C hapter X; W. Beckner, H. Kalf,
R. S. Phillips, and A. S. W ightm an for critically reading one or more
sections.
N um erous other colleagues for contributing valuable suggestions.
F. A rm strong for typing m ost of the prelim inary manuscript.
J. H agadorn, R. Israel, and R. W olpert for helping us with the proofreading.
Academic Press for its aid and patience; the N ational Science and Alfred
P. Sloan F oundations for financial support.
Jackie and M artha for their encouragement and understanding.


Introduction

A fu n ctio n a l an a lyst is an analyst, firs i and fo re m o si, and not a degeneróte species o f topologist.

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E. H ille

M ost texts in functional analysis suffer from a serious defect that is shared
to an extent by Volume I of M ethods of M odern M athem atical Physics.
Namely, the subject is presented as an abstract, elegant Corpus generally
divorced from applications. Consequently, the students who learn from these
texts are ignorant of the fact that alm ost all deep ideas in functional analysis
have their ímmediate roots in “ applications,” either to ciassical areas of
analysis such as harmonio analysis or pardal differential equations, or to
another science, prim arily physics. F or example, it was ciassical electromagnetic potential theory that m otivated Fredholm ’s work on integral
equations and thereby the w ork o f Hilbert, Schmidt, Weyl, and Riesz on the
abstractions of Hilbert space and com pact operator theory. And it was the
Ímpetus of quantum mechanics that led von Neum ann to his development
of unbounded operators and later to his work on operator algebras.
M ore deleterious than historical ignorance is the fact th at students
are too often misled into believing that the most profitable directions for
research in functional analysis are the abstract ones. In our opinión, exactly
the opposite is true. We do not m ean to imply that abstraction has no
role to play. Indeed, it has the critical role of taking an idea from a
concrete situation and, by elim inating the extraneous notions, m aking the
idea more easily understood as well as applicable to a broader range of


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situations. But it is the study of specific applications and the consequent
generalizations th at have been the m ore im portant, rather than the consideration of abstract questions about abstract objects for their own sake.
This volume contains a m ixture of abstract results and applications, while
the next contains mainly applications. The intention is to offer the readers
of the whole series a properly balanced view.
We hope that this volume will serve several purposes: to provide an
introduction for gradúate students not previously acquainted with the
material, to serve as a reference for m athem atical physicists already working
in the field, and to provide an introduction to various advanced topics
which are difficult to understand in the literature. N ot all the techniques
and applications are treated in the same depth. In general, we give a very
thorough discussion of the m athem atical techniques and applications in
quantum mechanics, but provide only an introduction to the problems
arising in quantum field theory, classical mechanics, and partial differential
equations. Finally, some of the material developed in this volume will not
find application until Volume III. F or all these reasons, this volume contains
a great variety of subject matter. To help the reader select which material
is im portant for him, we have provided a “ Reader’s G uide” at the end
of each chapter.
As in Volume I, each chapter contains a section o f notes. The notes
give references to the literature and sometimes extend the discussion in the
text. Historical comments are always limited by the knowledge and prejudices
of authors, but in m athem atics that arises directly from applications, the
problem of assigning credit is especialiy difficult. Typically, the history is
in two stages: first a specific m ethod (typically difficult, com putational,
and sometimes nonrigorous) is developed to handle a small class of problems.
Later it is recognized th at the m ethod contains ideas which can be used to
treat other problems, so the study of the m ethod itself becomes im portant.
The ideas are then abstracted, studied on the abstract level, and the
techniques systematized. W ith the newly developed m achinery the original

problem becomes an easy special case. In such a situation, it is often not
completely clear how m any o f the m athem atical ideas were already contained
in the original work. Further, how one assigns credit may depend on
whether one first learned the technique in the oíd com putational way or in
the new easier but m ore abstract way. In such situations, we hope that
the reader will treat the notes as an introduction to the literature and
not as a judgm ent of the historical valué o f the contributions in the papers
cited.
Each chapter ends with a set of problems. As in Volume I, parts of
proofs are occasionally left to the problems to encourage the reader to


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particípate in the development of the mathematics. Problems that fill gaps
in the text are m arked with a dagger. Difficult problems are m arked with
an asterisk. W e strongly urge students to work the problems since that
is the best way to learn mathematics.


Contents

IX :

T H E F O U R IE R T R A N S F O R M

1. The Fourier transform on SP(Un) and £P'(Un), convolutions
2. The range o f the Fourier transform: Classical spaces
3. The range o f the Fourier transform: Analyticity
4. I I Estimates

Appendix Abstract interpolation
5. Fundamental Solutions o f partial dijferential equations
with constant coefficients
6. Elliptic regularity
7. The free Hamiltonianfor nonrelativistic quantum
mechanics
8. The Gdrding-Wightman axioms
Appendix Lorentz invariant measures
9. Restriction to submanifolds
10. Products o f distributions, wave front sets, and oscillatory
integráis
Notes
Problems
Reader's Guide

vii
ix
xv

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Preface
Introduction
Contents o f Other Volumes

1
9
15
21
32

45
49

54
61
12
16
81
108
120
133


S E L F -A D JO IN T N E S S AND T H E E X IS T E N C E O F D Y N A M IC S

1. Extensions o f symmetric operators
Appendix Motion on a half-line, limit point-limit circle
methods
2. Perturbations o f self-adjoint operators
3. Positivity and self-adjointness I : Quadratic forms
4. Positivity and self-adjointness II: Pointwisepositivity
5. The commutator theorem
6. Analytic vectors
7. Free quantum fields
Appendix The Weyl relations for the free field
8. Semigroups and their generators
9. Hypercontractive semigroups
10. Graph Limits
11. The Feynman-Kac formula
12. Time-dependent Hamiltonians

13. Classical nonlinear wave equations
14. The Hilbert space approach to classicalmechanics
Notes
Problems
Reader’s Guide

¡35
146
162
176
182
191
200
207
231
235
258
268
274
282
293
313
318
338
349

List o f Symbols
Index

353

355

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X:


Contents of Other Volumes

Functional A nalysis

1 Preliminaries
II Hilbert Spaces
I I I Banach Spaces
I V Topological Spaces
V Locally Convex Spaces
VI Bounded Operators
VII The Spectral Theorem
V III Unbounded Operators

Volum e III:

XI
X II
X III

A nalysis of O perators

Perturbations o f Point Spectra
Scattering Theory

Spectral Analysis

Later Volum es

X IV
XV
XVI
X V II
X V III
X IX
XX

Group Representations
Commutative Banach Algebras
Convex Sets
The G N S Construction
Von Neumann Algebras
Applications to Quantum Field Theory
Applications to Statistical Mechanics

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Volum e 1:


IX :

The Fourier Transform

W e have therefore the equation o f condition

F (x) =

j dqQ

eos q x

IX.1

T h e Fourier transform on «^(or) and y"(lir),convolutions

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¡ f we substitu ted fo r Q any fu n ctio n o f q, and conducted the integration fro m q = 0 to q = oo,
we should fin d a fu n ctio n o f x : it is required to solve the inverse problem, that is to say, to
ascertain w hat fu n ctio n o f q, a fter being substituted fo r Q, gives as a resull the fu n ctio n F (x),
a rem arkable problem whose solution demands attentive examination.
Joseph Fourier

The Fourier transform is an im portant tool of both classical and modern
analysis. We begin by defining it, and the inverse transform, on 9 ’{Un),
the Schwartz space of C00 functions of rapid decrease.
D e fin itio n
/ given by

Suppose f e £f(Un). The Fourier transform o f / i s the function

where x • X = Y j =i x¡ ■The inverse Fourier transform of /, denoted by
f is the function

We will occasionally w r ite / = J 5/

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Since every function in Schwartz space is in ¿ '(R ”), the above integráis
make sense. M any authors begin by discussing the Fourier transform on
¿'(R"). We start with Schwartz space for two reasons: First, the Fourier
transform is a one-to-one m ap of Schwartz space onto itself (Theorem IX. 1).
This makes it particularly easy to talk about the inverse Fourier transform,
which of course turns out to be the inverse map. T hat is, on Schwartz
space, it is possible to deal with the transform and the inverse transform
on an equal footing. Though this is also true for the Fourier transform on
L2(R") (see Theorem IX.6), it is not possible to define the Fourier transform
on L2(R") directly by the integral formula since L2(R”) functions may not
be in I}(Un); a limiting procedure must be used. Secondly, once we know
that the Fourier transform is a one-to-one, bounded m ap of ^ (R " ) onto
£ f(ñ n), we can easily extend it to Sf'(W). It is this extensión that is funda­
mental to the applications in Sections 5, 6, and 8.
We will use the standard multi-index notation. A multi-index
is an n-tuple of nonnegative integers. The collection of all multi-indices
will be denoted by / + . The symbols |oc|, x*, D“, and x 2 are defined as follows:
n

dx«‘ dx“>■■■d x f
n

In preparation for the proof that

and ~ are inverses, we prove:

L e m m a The maps
and
are continuous linear transform ations of
■^(R") into ^(R "). Furtherm ore, if a and p are multi-indices, then
(IX.1)
Proof The m ap

is clearly linear. Since


We conclude that
l l / I U = sup| W ) ( A ) | < (2A / 2 |- m x » f ) \ dx < X
so takes .y (R ”) into .‘/"(R"), and we have also proven (IX. 1). Furtherm ore,
if k is large enough, J (1 + x 2)- * dx < co so that

~

dX

^ + x2)~'‘ í/x) S“ P{(1 + x2) +k| D* ( - ^ ) í/(^ )|}

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II/II-.A ^ ( ^ P Í r . (1 + X2)"*1

Using Leibnitz’s rule we easily conclude that there exist multi-indices ay,
Pj and constants c¡ so that
l l / I L ^ 1 ^ 1 1 /1 1 ^
¡=i
Thus,

is bounded and by Theorem V.4 is therefore continuous. The
proof for * is the same. |

We are now ready to prove the Fourier inversión theorem. The proof
we give uses the original idea of Fourier.
T h e o r e m IX.1 (Fourier inversión theorem )
The Fourier transform is a
linear bicontinuous bijection from ^ ( R ”) onto £f(R n). Its inverse m ap is
the inverse Fourier transform, i . e . , / = f = J .
X

9

5

Proof We will prove that f —f The proof th at/ = / is s im ila r./= / implies
A
X
A
v
that is surjective and / = / implies that
is injective. Since and
are
continuous maps of £ f ( ñ n) into ^ ( R ”), it is sufficient to prove that / = /
for / contained in the dense set CJ(R"). Let Ce be the cube of volume
(2/e)” centered at the origin in R". Choose e small enough so that the
support o f/ is contained in C£. Let
K e = {k e R |each kjne. is an integer}
Then
/ ( x ) = Z ({ ¥ Y n eik'

keK,

f ) ( W 2eik ‘*

is just the Fourier series o f/ which converges uniformly in C£ t o / s in c e /is
continuously differentiable (Theorem II.8). Thus
/ W

=

E

w

k e K, ( ¿ n )

W

<IX 2 >


Since IR" is the disjoint unión of the cubes of volume (rce)" centered about
the points in K c, the right-hand side of (IX.2) is just a Riemann sum for
the integral of the function/ ( k)e‘k ' x/(2n)"12. By the le m m a ,/(k)eik' x e ¡S(IR"),
so the Riemann sums converge to the integral. T h u s/ = / |
C o ro llary

Suppose / e y (IR"). Then

j

f | / ( x ) |2 dx = [ \ f ( k ) \ 2 dk
R.

j R.

/ ( * ) = E ({ h Y l2eik' x, f ( x ) ) ( h f ' 2eik' *
ke Kt

Since {(£e)"l2eik ' x}ke k, is an orthonorm al basis for L2(Ce),
[ |/ ( x ) [ 2 dx = ¡ \ f ( x ) \ 2 dx
J W"
J c,
= E \ ( ( W l2eik x, f ( x ) ) l 2
k e K,

=

e
k e K,

i/w iv r

-7Z Z *¡ | f ( k ) \ 2 dk
J R"

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Proof This is really a corollary of the proof rather than the statem ent of
Theorem IX. 1. If/ has com pact support, then for e small enough,

This proves the corollary for f e Cq. Since and |¡ • ¡|2 are continuous on
í f and Cq is dense, the result holds for all of Sf. |
E x a m p le 1
where a > 0.

We compute the Fourier transform o ff ( x ) = e~ ax2/2 e S?(M)

f(X) =

=

í e -«x2/2e - a - x dx
Jr
f fe exp( —í2 — itX f e \ dt
uy/Eñ
2n J r v a
\
V a/
e ~ k2/2lx r
/
X \2
—^
exp - í + ¡ — —
y j an
\
y/2a,
e ~ X2/2x f

an

e ' dj t

e ~ X2fla


The next to last step follows from the Cauchy integral formula and the
exponential decrease of e ~ z* along lines parallel to the x axis.
We now define the Fourier transform on .9"(Un).
D e fin itio n
Let T e .5^'(R"). Then the Fourier transform of T, denoted by
í , is the tempered distribution defined by T(
Tá
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Suppose that h, (pe ■9?(Un), then by the polarization identity and the
corollary to Theorem IX. 1 we ha ve (h, q>) = (/i, we obtain

where and Tt are the distributions corresponding to the functions g and g
respectively. This shows that the F ourier transform on y ”(R") extends the
transform we previously defined on .5^(R").

T h e o r e m IX.2
The Fourier transform is a one-to-one linear bijection
from y"(R ") to ^ '(R " ) which is the unique weakly continuous extensión of
the Fourier transform on £/>(Un).
Proof If (p„ -* T(cp) for each T

in .9"(Un). Thus T(q>„) -» T(
functional on ^ (R "). Furtherm ore, if T„^*T, then T„^*T because
Tn((j>) -* T((p) implies Tn((p) -* T{(p). Thus Ti-* T is weakly continuous.
The remaining properties of follow immediately from the correspond­
ing statem ents on ^ (R " ) (see Problem 19 in C hapter V). |
E x a m p le 2
We com pute the Fourier transform of the derivative of the
delta function at b e R :
S'„(
" á‘( ( S p f

■ /(

t

í

H

*

So, the Fourier transform of S'b is the function ixe - ¡bx/ ^ 2 n -


We now introduce a new operation on functions.
D e fin itio n s
Suppose th a tf g e ■9>(Un). Then the convolution o f/ and g,
denoted by f * g, is the function

T h e o re m IX.3
(a)

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The convolution arises in many circumstances (we have already used it in
discussing closed operators in Section VIII. 1). In Section 4 we use interpolation theorems to prove U estimates on the convolution f * g in terms of
/ and g. In this section we concéntrate on the properties of the convolution
as a m ap from SP(lR") x £f(W') to ■9*(Un). Using these properties we show
that the convolution can be extended to a m ap from ■9í'(Mn) x
to
0 nM , the polynomially bounded C°° functions. Convolutions frequently
occur when one uses the Fourier transform because the Fourier transform
takes products into convolutions (Theorem IX.3b and Theorem IX.4c).

For each / e SP{R"), g - > f * g is a continuous m ap o f SP{UR) into

y ( R a).
(b) fg = (2n)~nj2f * g and f T g = (2n)nl2fg.
(c) F or f, g, h, in ^(OT), f * g = g * f and / * (g * h) = ( / * g) * h.

Proof From the polarization identity and the corollary to Theorem IX. 1
we find that (apply this identity to eiy '*f(x) and g obtaining (e,y xf, g) = (ely xJ, g). But

and
g — iX ■x + i y ■x

e a x+,y * f ( x ) d x U(A) dX

R" \

J R"

R”

which proves that fg — (2n) nl2f * g. Using the inverse Fourier transform
this formula may be stated as
(2n)nl2f g - = f * g


This shows that convolution is the composition of the inverse Fourier
transform, m ultiplication by (2n)nl2f, and the Fourier transform. It follows
that convolution is continuous.
The statem ents in (c) follow trivially from (b). |

D e fin itio n
Suppose that f e £P(U"), T e &”(R") and let / ( x ) denote the
function,/ ( —x). Then, the convolution of T and / denoted T * f is the
distribution in £P’(U") given by
( T * f ) ( c p ) = T ( J *cp)
for all

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In order to extend the m ap C g - * f *g to 9 ”, we look for a continuous
m ap
: Sf -* S f so that C'f f1 — Cf . We then define C'f to be convolution
on ¡T.

The fact that g - » / * g is a continuous transform ation guarantees that
T * f e £P'(W). The following theorem summarizes the properties of this
extended convolution.
Let f y denote the function f y(x) —f ( x — y) and Jy the function f ( y — x).
W h e n /is given by a large expression (•••), we will sometimes write (•••)'
rather than (•'•).
T h e o re m IX,4
For each f e .9’(Un) the m ap T -* T * f is a weakly
continuous m ap of
into
( ñ tt) which extends the convolution on
^(R "). Furtherm ore,
(a)

T * f is a polynomially bounded C x function, i.e. T * f e On
M . In fact,
{T * f) ( y ) = T(?y) and
D^(T * f ) = (D^T) * f = T *

(b)

(T * f)* g = T * (f* g )

(c)

r T f = ( 2 n f'2f t

(IX.3)

Proof Since T -> T * f is defined as the adjoint of a bounded m ap from .cf

to y , it is autom atically weakly continuous. The fact that it extends the
convolution on Sf is just a change of variables. The statements (IX.3), (b),
and (c), all follow immediately from the corresponding statements for T e
and the facts that í f is weakly dense in 9 " and that
, D multiplication
b y /, and convolution are all weakly continuous on .9".


It remains to prove the first part of (a). Since T e .9?'(Un), it follows
from the regularity theorem (Theorem V.10) that there is a bounded
continuous function h, a positive integer r, and a multi-index /? so that
T (? y ) =

jf R.M

* )(l

+

^ Y (D ^ f){y

~

* ) d x

| T ( 7 , ) | < ||/ i |L

f

( l + x 2Y \ ( D ' f ) l y - x ) \ d x

J R"

= 11*11. í (l + ( y - T ) 2r |D y (T )|d T
J R"

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Since D^f e Sf, T (Jy) is an infinitely differentiable function of y. The change
of variables i = y — x shows that

from which it follows easily that y*-* T(Jy) is polynomially bounded.
A similar proof works for the derivatives of jn-» T (Jy). Thus T (Jy) e 0"M .
Suppose that a distribution S e Sf'{W ) is given by a polynomially
bounded continuous function s. Then, using Fubini’s theorem we find that
for q>6 Sf(MR)
(S *f)(cp) = S(J*cp)
= | s ( x ) |j / ( x - y)
= j ( | s(x)7(x)dxj</>(y) dy
= w i m

so S * / = S(Jy). By the regularity theorem T = D“S for some such S. Thus
by (IX.3)
T * f = (D « S )* f = S*D *f
= S(W )~y)

= ( —i p s m D )
= i r s ( ? y)
= n i)

This completes the proof. |


T h e o re m IX .5

Let T e SP'(U”) and / e

Then / T e 0 M
n and

f T ( k ) = (2 n Y " n T ( f e ~ * *). In particular, if T has com pact support and
ijf e
is identically one on a neighborhood o f the support of T, then

t[k) =

{2n)-nl2T{\pe-i k x )

Proof By Theorem IX.4c and the Fourier inversión formula we have
f f = (2n)~nl2J * T. T h u s / T e 0 nM and

= {2n)-”l2T ( e - ik *f) |

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ff(k) = (2ti)-"l2T{?k)
We rem ark th at one can also define the convolution of a distribution
T e ^ '(R " ) with an / e í^(RB) by (T * f ) ( y ) = T (Jy). A proof sim ilar to the
p roof o f Theorem IX.4 shows that T * / i s a (not necessarily polynomially
bounded) C® function and that (IX.3) holds.

We have already introduced the term “ approxim ate identity” in Section
VIII. 1; we now define it formally.

D e fin itio n
Let j(x) be a positive C® function whose support lies in the
sphere of radius one about the origin in R" and which satisfies | j{x) dx = 1.
The sequence of functions j c(x) = e~nj(x/e) is called an approximate identity.
P rop ositio n

Suppose T e .9”(U") and let j c(x) be an approxim ate iden­
tity. Then T * j e - * T weakly as e -* 0.

Proof If q>e ^ ( R ”), then (T * j t){(p) = T(JC*
that j t * (p- -l - 'U (p. To do this it is sufficient to show that (2n)nl2j c(p cp.
Since j e(X) = j(eX) and /(O) = (2n)~nl2, it follows that (2n)ttl2j c(x) converges
to 1 uniformly on com pact sets and is uniformly bounded. Similarly,
rpjc converges uniformly to zero. We conclude that (2n)nl2Jt

IX.2 The range of the Fourier transform: Classical spaces

We have defined the Fourier transform on SP(W) and 6P'(W). In this
section, Section IX.3, and Section IX.9, we investígate the range of the
F ourier transform when it is restricted to various subsets of £P'(U"). These


questions are natural and have historical interest, but more im portant,
characterizing the range of the Fourier transform is very useful. One is often
able to obtain inform ation about the Fourier transform of a function and
one would like to know what this says about the function itself. We begin

with two theorems which follow easily from the work that we have already
done in Section IX. 1.

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T h e o re m IX.6 (the Plancherel theorem)
The Fourier transform extends
uniquely to a unitary m ap of L2(R") onto L2(R"). The inverse transform
extends uniquely to its adjoint.

Proof The corollary to Theorem IX.1 States that if / e
then
| | / | | 2 = | | / | | 2 . Since J r [ y >] = £P, 3F is a surjective isometry on L2(R"). |

T h e o re m IX.7 (the Riem ann-Lebesgue lemma)
The Fourier transform
extends uniquely to a bounded m ap from L1(Rn) into C ^ R "), the continuous
functions vanishing at oo.
Proof For / 6 SP{U"), we know that / e
estímate

and thus / e Cc0(R”). The

is trivial. The Fourier transform is thus a bounded linear m ap from a
dense set of 1}(W) into C ^ R "). By the B.L.T. theorem,
extends
uniquely to a bounded linear transform ation of ¿ (R " ) into C00(R"). |

We rem ark that the Fourier transform takes U (R") into, but not onto
C00(R") (Problem 16).

A simple argument with test functions shows that the extended transform
on L^R") and L2(R") is the restriction of the transform on ^ '(R "), but
it is useful to have an explicit integral representation. F or / 6 L^R"), this
is easy since we can find f m e SP(W) so that | | / — f m\\ i -> 0. Then, for
each X,
/(A ) = lim (/m(A))


So, the F ourier transform of a function in 1}(W) is given by the usual
formula.
Next, suppose / e L2(R") and let

X gl

'

|0

Then %R f e l ) (R") and ~/R f • £

>f

so by the Plancherel theorem

R^ x »f. F o r Xr f we have the usual formula; thus
/( A ) = l.i.m .(27t)~n/2 [
R-ao

e ~ ,Xxf ( x ) d x

|x| < /?

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XR f

|* | > R

where by “ l.i.m.” we mean the limit in the L2-norm. Sometimes we will
dispense with ¡x| ^ R and just write
/ ( A ) = l.i.m .(27t)-n/2 J e ~ ,X xf ( x ) dx

for functions / e L2(R").
We have proven above that L2(R ")-►L2(R") and L ^ R ")-►L00(Rn) and
in both cases is a bounded operator. It is exactly in, situations like this
that one can use the interpolation theorems which we will prove in the
Appendix to Section 4.

T h eo rem IX .8 (Hausdorff-Young inequality)
Suppose 1 < q < 2, and
p _1 + q ~ l — 1. Then the Fourier transform is a bounded m ap of Í?(R")
to Zf(R") and its norm is less than or equal to (2n)n{ll2~ 1/9).

Proof We use the Riesz-Thorin theorem (Theorem IX. 17) with q0 = 2 =
p0 , p, = co, and q t = 1. Since | | / | | 2 = | | / | | 2 and | | / | | „ á (27c)"n/2|) / ||„ we
conclude th at | | / | | Pi < Ct\\f\\qi where p, 1 = (1 - í)/2, q, 1 = (1 - t)/2 + t =
1 — p,
and log C, = í log(27c)_n/2. |

W e now come to another natural question. W hat are the Fourier transforms of the finite positive measures on R"? Suppose th at we define

Then, if (pe
fi(X)(p(X) dX = (2n) n/2 [ [ [

e lX x dfi(x)\(p(X) dX

= (27e)~"/2 í [ í

J R“\ j

e ,x' x(p{X) dX j dfi(x)
R"

so this definition coincides with the restriction of the Fourier transform
on y { U n) to the positive measures. Suppose L,,
XN e U n and % =

This shows that the function fi(k) has the property that for any Lj,
k N e IR", {/¿(>-¡ — Xj)} is the matrix of a positive operator on CN. Furtherm ore,
by the dominated convergence theorem, ft is continuous, and since
|/i(A)| < (27r)-^2 f \e~iX-x \d n (x)
J IR "

= (27r)"',/2/i(Rn)
£(•) is also bounded.

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q>(x) dfi(x)

D efin itio n
A complex-valued, bounded, continuous function/ on IR" that
has the property that { /(L ; - L,)}; ¡ is a positive matrix on C* for each N
and all Lt , . .., XN e IR" is called a function of positive type.

There are three properties of functions of positive type which follow
easily from the definition. Letting N — 1, x e IRW,
(1)

/(0 )> 0

since /(O) is a positive operator on C . Letting N = 2, and choosing
X¡ = x, X2 = 0, we see that the matrix

m

m

/ ( - x ) /(O),
must be positive and therefore self-adjoint with positive determinant. This
implies that
(2)
(3)

f(x ) = f ( - x )
|/ ( x ) ¡ < / ( 0 )


Notice th at in proving these three properties we did not use the fact that

f ( x ) is bounded, so we could have left out the word bounded in the
definition and recovered boundedness from (3) above. It is clear that any
convex com binations or scalar m últiples of functions of positive type again
give functions of positive type, so these functions form a cone.
T h e o r e m IX.9 (Bochner’s theorem )
The set of Fourier transform s of
the finite, positive measures on IR" is exactly the cone of functions of
positive type.

(
I

/ ( x - y)i/r(x)
x, y s R”

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Proof W e do not give Bochner’s original proof but rather an easy,
interesting argum ent based on Stone’s theorem. We have already shown
that the Fourier transforms of finite positive measures are functions of
positive type. W e need to prove the converse. S u p p o se /is of positive type.
Let X denote the set of complex-valued functions on R" which vanish
except at a finite num ber of points. Then

has all the properties of a well-defined inner product except that we may
have (, then
X / X is a well-defined pre-Hilbert space under (•, •)/ . Suppose that t e R"
and define Ut on X by (Ut (p)(x) =

(•, -)j-, it takes equivalence classes into equivalence classes and thus
restricts to an isometry on X / X . Since the same is true of
this
isometry has dense range and thus extends to a unitary operator 0 , on
X = X / X . Furtherm ore, 0 t+ s = ¿7, 0 S , Ü0 = /, and because o f the continuity of f 0 , is strongly continuous. Thus the m ap t -> 0 , satisfies the
hypotheses of Theorem VIII. 12 (the generalization of Stone’s theorem).
Therefore, there is a projection-valued measure
, on IR" so that
(<P>Üt'l')f=\i r e lt' x d(q>,P^\¡f)s
¡s
j m»
Let q>0 denote the equivalence
:nce class containing the function
jl,
(p0{x) = |0,

x= 0
x 0

Then
/ ( t ) = Üt 0)f
so we have displayed /
measure. |

as the Fourier transform of a finite positive