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

We begin by defining it, and the inverse transform, on 9 ’{Un, the Schwartz space of C 00 functions of rapid decrease... We start with Schwartz space for two reasons: First, the F ourie

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H e le n and Gerald R e e d

M in n ie and H y Sim ón

This volum e continúes our series of texts devoted to functional analysis

m ethods 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 C hapters IX and X ;

we expect th at 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 m ethods

It gives us pleasure to thank m any 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 m ore 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 proof- reading

Academic Press for its aid and patience; the N ational Science and Alfred

P Sloan F oundations for financial support

Jackie and M arth a for their encouragem ent and understanding

M ire R eed

June 1975

A fu n c tio n a l a n a lyst is an analyst, fir s i and fo re m o si, and no t a degeneróte species o f topologist.

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 ro ots in “ applications,” either to ciassical areas of

analysis such as harm onio analysis or p ardal differential equations, or to another science, prim arily physics F o r example, it was ciassical electro- magnetic potential theory that m otivated F redholm ’s work on integral equations and thereby the w ork o f Hilbert, Schmidt, Weyl, and Riesz on the abstractions of H ilbert space and com pact operator theory And it was the Ímpetus of quantum mechanics th at led von N eum 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 a t students are too often misled into believing th at the m ost 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 th at 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 m ore easily understood as well as applicable to a bro ad er range of

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 considera- tion of abstract questions about abstract objects for their own sake.

This volume contains a m ixture o f 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 o t 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, bu t provide only an introduction to the problem s arising in quantum field theory, classical mechanics, an d partial differential equations Finally, som e of the m aterial developed in this volume will not find application until Volume III F or all these reasons, this volume contains

a great variety of subject m atter T o help the reader select which m aterial

is im portant for him, we have provided a “ Reader’s G u id e” 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 H istorical comm ents 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 o f problem s 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 F urther, 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 th at 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

particípate in the development of the mathematics Problem s 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 problem s since that

is the best way to learn mathematics

5 Fundamental Solutions o f partial dijferential equations

7 The free Hamiltonianfor nonrelativistic quantum

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

Appendix Motion on a half-line, limit point-limit circle

Contents of Other Volumes

V o lum e 1: Fu n ctio n al A n a ly sis

X I Perturbations o f Point Spectra

IX : The Fourier Transform

W e have therefore the equation o f condition

¡ f we su b stitu ted fo r Q a n y fu n c tio n o f q, and conducted the integration fro m q = 0 to q = oo,

we should fin d a fu n c tio n o f x : it is required to solve the inverse problem , th a t is to say, to ascertain w hat fu n c tio n o f q, a fter being su b stitu ted fo r Q , gives as a resull the fu n c tio n F (x),

IX.1 T h e Fourier transform on «^(or) and y"(lir),convolutions

The F ourier transform is an im portant tool of both classical and m odern

analysis We begin by defining it, and the inverse transform, on 9 ’{Un),

the Schwartz space of C 00 functions of rapid decrease

D e fin itio n Suppose f e £f(Un) The Fourier transform o f / i s the function

Since every function in Schwartz space is in ¿ '(R ”), the above integráis make sense M any authors begin by discussing the F ourier transform on

¿'(R") We start with Schwartz space for two reasons: First, the F ourier 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 F ourier 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 F ourier transform

on L2(R") directly by the integral formula since L2(R”) functions may not

be in I}(Un); a limiting procedure m ust be used Secondly, once we know

that the Fourier transform is a one-to-one, bounded m ap o f ^ (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:

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

n

dx«‘ dx“> ■ ■ ■ d x f n

(IX.1)

Proof The m ap is clearly linear Since

We conclude that

l l / I U = su p| 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

II/II-.A ^ ( ^ P Í r (1 + X2)"*1 dX

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

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

We are now ready to prove the F ourier 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 F ourier 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

Proof We will prove that f —f The proof th a t/ = / is s im ila r /= / implies

that is surjective and / = / implies that is injective Since and are

continuous m aps 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 |eac h kjne is an integer}

Then

/ ( x ) = Z ( { ¥ Y n eik' f ) ( W 2eik ‘ * keK,

is ju st the F ourier series o f/ which converges uniformly in C£ t o / s in c e /is continuously differentiable (Theorem II.8) Thus

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 ju st 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 lla ry Suppose / e y (IR") Then

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

Proof This is really a corollary o f 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 We com pute the F ourier transform o ff ( x ) = e ~ ax2/2 e S?(M)

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 F ourier transform on 9"(Un).

D e fin itio n Let T e 5^'(R") Then the F ourier transform of T, denoted by

í , is the tem pered distribution defined by T(<p) = T(<p).

Suppose that h, (pe ■9?(Un), then by the polarization identity and the corollary to Theorem IX 1 we ha ve (h, q>) = (/i, <p) Substituting g = g for h,

we obtain

Tá<p) = f f a M * ) dx = f 0(*)<P(x) dx = Tt {<p) =

where and Tt are the distributions corresponding to the functions g and g

respectively This shows th at 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 F ourier transform is a one-to-one linear bijection from y"(R ") to ^ '( R " ) which is the unique weakly continuous extensión of

the F ourier transform on £/>(Un).

Proof If (p„ -* <p, then by Theorem IX 1, cpn cp, so T(cpn) -> T(cp) for each T

in 9"(Un) Thus T(q>„) -» T(<p), which shows that T is a continuous linear 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 correspond­ing statem ents on ^ (R " ) (see Problem 19 in C hapter V) |

E x a m p le 2 We com pute the F ourier transform of the derivative of the

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

The convolution arises in many circumstances (we have already used it in discussing closed operators in Section VIII 1) In Section 4 we use interpola-

tion theorems to prove U estim ates 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 n M, the polynomially bounded C°° functions C onvolutions frequently

occur when one uses the F ourier transform because the F ourier transform takes products into convolutions (Theorem IX.3b and Theorem IX.4c)

Proof From the polarization identity and the corollary to Theorem IX 1

we find that (<p, i//) = ((p, í¡t) for <p, tp e £P(M") Letting y e IR" be fixed, we apply this identity to eiy '*f(x) and g obtaining (e,y xf , g) = (ely xJ, g) But

which proves that f g — (2n) nl2f * g Using the inverse F ourier transform

this formula may be stated as

(2n)nl2f g - = f * g

This shows that convolution is the com position 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) |

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.

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 <p e £P(Un).

The fact that g - » / * g is a continuous transform ation guarantees that

T * f e £P'(W) The following theorem summarizes the properties o f 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 r e m IX,4 F or 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

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 S f is just a change of variables The statem ents (IX.3), (b), and (c), all follow immediately from the corresponding statem ents 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 ) = f M * ) ( l + ^ Y ( D ^ f ) { y ~ * ) d x

j R.

Since D^f e Sf, T (J y) is an infinitely differentiable function of y The change

of variables i = y — x shows that

| T ( 7 , ) | < ||/ i |L f ( l + x 2 Y \ ( D ' f ) l y - x ) \ d x

J R"

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

J R"

from which it follows easily that y*-* T (Jy) is polynom ially bounded

A sim ilar proof works for the derivatives of jn-» T (J y) Thus T (J y) e 0"M Suppose that a distribution S e S f'{W ) is given by a polynomially

bounded continuous function s Then, using F ubini’s theorem we find that

T h e o re m IX 5 Let T e SP'(U”) and / e Then / T e 0 n M 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 o f T, then

t[k) = {2n)-nl2T {\pe-i k x ) Proof By Theorem IX.4c and the Fourier inversión form ula we have

f f = (2n)~nl2J * T T h u s / T e 0 n M and

ff(k) = (2t i)-"l 2 T{?k)

= {2n)-”l2T ( e - ik *f) |

We rem ark th at one can also define the convolution o f a distribution

T e ^ '(R " ) with an / e í^(R B) by (T * f ) ( y ) = T (Jy) A proof sim ilar to the

p ro o f 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 SectionVIII 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 approxim ate identity.

P ro p o sitio 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 * <p), so it is sufficient to show

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 <p ^+<p §

IX.2 Th e range of the Fourier transform: Classical spaces

We have defined the F ourier 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 m ore im portant, characterizing the range of the F ourier transform is very useful O ne is often able to obtain inform ation about the F ourier 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.

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

| | / | | 2 = | | / | | 2 Since J r [ y >] = £P, 3F is a surjective isometry on L2(R") |

T h e o r e m IX.7 (the R iem 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 F or / 6 SP{U"), we know that / e and thus / e C c0(R”) Theestímate

is trivial The F ourier transform is thus a bounded linear m ap from a

dense set o f 1}(W) into C ^ R " ) By the B.L.T theorem, extends uniquely to a bounded linear transform ation of ¿ (R " ) into C 00(R") |

We rem ark that the F ourier transform takes U (R") into, but not onto

C 00(R") (Problem 16)

A simple argum ent 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 ou rier transform o f a function in 1}(W) is given by the usual

formula

Next, suppose / e L2(R") and let

X g l ' | 0 | * | > R

Then %R f e l ) (R") and ~/R f • £ >f so by the Plancherel theorem

XR f R^ x »f F o r X r f we have the usual form ula; thus

/ ( A ) = l.i.m (2 7 t)~ n/2 [ e ~ ,Xxf ( x ) d x

where by “ l.i.m.” we m ean the limit in the L2-norm Sometimes we will

dispense with ¡x| ^ R and ju st write

/ ( A ) = l.i.m (27t)-n /2 J e ~ ,X xf ( x ) dx

for functions / e L2(R")

We have proven above th at 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 theorem s which we will prove in the Appendix to Section 4

T h e o re m IX 8 (H ausdorff-Y oung inequality) Suppose 1 < q < 2, and

p _1 + q ~ l — 1 Then the F ourier transform is a bounded m ap of Í?(R")

to Zf(R") and its norm is less than o r equal to (2n)n{ll2~ 1/9).

Proof We use the R iesz-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 F ourier trans- forms of the finite positive measures on R"? Suppose th at we define

so this definition coincides with the restriction of the F ourier 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 C N Furtherm ore,

by the dom inated convergence theorem, ft is continuous, and since

£(•) is also bounded

D e fin itio n A complex-valued, bounded, continuous function/ on IR" that

has the property that { /(L ; - L,)}; ¡ is a positive m atrix on C* for each N and all L t , , 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,

since /(O) is a positive operator on C Letting N = 2, and choosing

X¡ = x, X2 = 0, we see that the matrix

must be positive and therefore self-adjoint with positive determ inant This implies that

|/i(A)| < (27r)-^2 f \e~ iX-x \d n (x )

N otice 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 w ord 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 F ourier transform s of the finite, positive measures on IR" is exactly the cone o f functions of positive type

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 F ourier transform s of finite positive measures are functions o f positive type W e need to prove the converse S u p p o s e /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

(<l',<p)f = I / ( x - y)i/r(x)<p(y)

x, y s R”

has all the properties of a well-defined inner product except th at we may

have (<p, <p)f = 0 for some <p # 0 If we let J í be the set o f such q>, then

X / X is a well-defined pre-H ilbert space under (•, •)/ Suppose that t e R"

and define Ut on X by (Ut (p)(x) = <p(x — t) Since Ut preserves the form (•, -)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 operato r 0 , on

X = X / X F urtherm ore, 0 t+ s = ¿7, 0 S , Ü0 = /, and because o f the con-

tinuity 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

The notion o f positive type may be generalized to distributions If f ( x )

is a bounded continuous function, then / (x) will be of positive type if and only if

J J /( * ~ y)<p(y)<P(x) d x d y > 0 (IX.4)

for all cp e C q (Rn) To see this one need only approxim ate the integral in (IX.4) by a Riem ann sum This condition can be rewritten as

| | / ( t ) < p ( x — z)cp(x) dz dx = | f{z)(q> * cp)(z) dz > 0 (IX.5)

where q> is the function ¿p(x) = cp( — x) This suggests the following definition.

D e fin itio n A distribution T e 3*'(Un) is said to be of positive type if T(<p * (p ) > 0 for all cp e S>(Rn).

The following generalization of Bochner’s theorem is due to Schwartz This theorem is particularly interesting since it implies that certain ordinary distributions must be tempered The proof is sketched in Problem 20 (or see the N otes for a reference)

T h e o r e m IX.10 (the B ochner-Schw artz theorem ) A distribution

T e 3?'(R") is a distribution of positive type if and only if T e Rn) and

T is the F ourier transform of a positive measure of at m ost polynom ial

growth

If / ( x ) is a function of positive type, then this theorem implies that the weak derivatives ( —A)”1/ are all distributions o f positive type F o r / = ¿i,

a finite measure by Theorem IX.9, and ( - A)m/ = | x a positive measure

of polynom ial growth

Finally, we determ ine which bounded m easurable functions are distribu­tions of positive type A bounded measurable function / on Rn is said to

be of weak positive type if (IX.4) holds Since (IX.5) follows from (IX.4), the distribution

7>(<P) = | f ( x )<P(x ) dx

is of positive type and therefore 7} = /i, a polynomially bounded positive

measure If j c(x) is an approxim ate identity that is symmetric abo ut the

origin, then

11/11 =0 > Tf {jE* j c) = Tf { { j ^ J c))

= (2nYl2fi(\je(x)\2)

= (2n f 12 j |j£(x )|2 dfi(x)

O n each com pact subset of IR", j £(x) converges uniformly to (2;t)- "/2 as

£-►■0, so the /i-m easure of any com pact set is less than (2n)nll\\f\\rXj, so

H is finite.

We now come to the interesting point Since ¡x is finite, its F ourier transform is a continuous function of positive type Since /x and / m ust

coincide a.e., we have proven:

P r o p o s itio n A bounded function of weak positive type is equal alm ost everywhere to a continuous function of positive type

IX.3 T h e range of the Fourier transform : A nalyticity

In this section we investígate the connection between the decay properties

o f a function o r distribution at infinity and the analyticity properties of its

F ourier transform The m ost extreme form of decay at infinity is to have com pact support We will prove the Paley-W iener and Schwartz theorem s which characterize explicitly the F ourier transform s o f C°° functions and distributions with com pact support We then state two theorem s relating exponential decay to analyticity properties of the F ourier transform We cióse the section by characterizing the F ourier transform s o f tem pered distributions whose supports lie in sym m etric cones There are m any other theorem s of this genre; some of them are discussed in the Notes

Suppose t h a t / e Cq (R")- Then for all ( = <C1( , £„> e C", the integral

/ ( ( ) = (2;t)~"/2 J e ~ K xf ( x ) d x

is well defined Furtherm ore, /( £ ) is an entire analytic function of the n

complex variables £2 , , since we can differentiate under the

integral sign In addition, if the support of / is contained in the sphere of

radius R, then an integration by parts yields

n w f t o = (2* y nl2 í e - * - * i r f ( x ) dx

Taking the absolute valué of both sides and using the fact th at / ( ( ) is

bounded on the set {£| |Im (\ < e}, we easily conclude that for each N,

1 /(0 1 < (1N+ -|^ | r for a lie e C"

where CN is a constant that depends o n N and f The interesting fact is

that these estimates are not only necessary but also sufficient for / to be in

Q ( R n).

T h e o r e m IX.11 (the Paley-W iener theorem ) An entire analytic function

of n complex variables g(Q is the F ourier transform of a Cq (IR") function with support in the ball (x| |x¡ < Rj if and only if for each N there is a

CN so that

for all C 6 C".

Proof We have already proven the “ only if ” part Suppose that g is entire

and satisfies estimates of the form (IX.6) Let f = X + ir¡, where A, rje R" Then for each r¡, g(A + ir¡) is in £P(Un) as a function of A, since the derivatives

fall off polynom ially by (IX.6) and the Cauchy formula Let

J R«

Then by Theorem I X l , / e £/’(Un) and g(A) = f(A) We want to show that

f ( x ) has support in the ball of radius R Because of the estimates (IX.6)

and Cauchy’s theorem, we can shift the región of integration in (IX.7) so that

/ ( x ) = (2n)~nl2 í eil* + i' ) xg(A + ir¡) dA (IX.8)

where we have chosen N large enough so that the integral on the right is finite N ow , / (x) does not depend on r¡, so if we let r¡ -> oo in an appropriate direction, we conclude that |/ ( x ) | = 0 if |x | > R |

This theorem has a natural generalization to the distributions with

com pact support Recall that a distribution T e 6P'(Un) has support in a closed set K if and only if T(q>) = 0 for every test function q> with support

in Un\K If K is compact, then T is said to have compact support The set

of distributions with com pact support is the dual space of (seeProblem s 39 and 40 of C hapter V)

T h e o r e m IX.12 A distribution T e £f'(R n) has com pact support if and only if T has an analytic continuation to an entire analytic function of n variables T (() that satisfies

|f( C ) | < C (! + |C |)A'e i' |Im!:| (IX.9)

for all C e C" and some constants C, N, R M oreover, if (IX.9) holds, the support of T is contained in the ball o f radius R.

Proof Suppose that T e £f”(Mn) has com pact support and let cp be a Cq (Rn)

function which is equal to one on the support of T Define F(C) =

T[(27t)~n/2e ~ i< X<p(x)] By Theorem IX.5, F(A + iO) is the F ourier transform

of T Furtherm ore, since

exp( — i(Xj(Cj + hj) + £ k * j Ck xk))<p(x) ~ e~ i( x<p(x)

hJ

y(B-)» — ixj e ~ K X(p(x) and T e £f”(W), F(£) is differentiable in the complex sense in each variable

and is thus entire

Since T e £T(Rn),

l«|SAÍl/»|SAÍ

for some N and C j and all / e y ( R ”) Thus, if <p has support in the sphere of radius R, then

|F(C)| < C 2(l + R n)(l + |f Conversely, suppose that F(Q is an entire function satisfying the estímate

(IX.9) T hen F(A + ¿0) e ^ '(R " ), so it is the F ourier transform of some

T e £ f'(R n) Let j\(x) be an approxim ate identity Then by Theorem IX.4,

T * j c = (2n)~"l2j c(Á)F(Á) Since j e has com pact support in {x| |x | < e}, we

know by the Paley-W iener theorem, that for each M we can find a constant CM so that

T * j t is contained in the sphere of radius R + e Since e is arbitrary and

{T * j c) -* T weakly, we conclude that the support of T is contained in the

sphere of radius R about the origin |

One natural way to extend the above theorem s is to replace “ com pact support ” with som e weaker notion of decay at infinity The following pair

of theorems (whose proofs are outlined in Problem 76) will be used in

C hapter XIII to prove the exponential decay of bound States of atom ic Hamiltonians

T h e o r e m IX.13 Let / b e i n L 2(IR'') Thene*’1*)/ e L2(IR'1) for allí? < a i f a n d

only if / has an analytic continuation to the set {£ j ¡Im f | < a} with the property that for each r¡ e U ” with \r¡\ < a, ?(■ + ir¡)e l}(Un) and for any

b < a, sup|„| <b ¡|T(- + i?/)||i < oo Then T is a bounded continuous function

and for any b < a, there is a constant Cb so that

| T ( x ) | ^ C t e - fcl*l

The next natural question to ask is w hat are the analyticity properties of

a function or distribution with support on a half-line, half-space, or in general in a cone? As a simple ex ampie, consider the F ourier transform / of

a function/ e 6f(U) which has support contained in [0, oo) The reader can

easily verify that

/(A - it]) = (2n ) ~ 112 J e - * k- * ) xf { x ) dx (IX.10)

is a well-defined analytic function in the open lower half-plane (i.e for rj > 0) and th at /(• — ir¡ ) - ^ * / a s r¡ J 0 T hat is, / , which need not be real analytic,

is the “ boundary valué” of an analytic function in the lower half-plane The study o f the F ourier transform s of functions and distributions with supports

in half-spaces dates back to the ciassical investigations of the Laplace transform and has played an im portant role in m odern analysis The main ideas and techniques are sim ilar to those used above in Theorem IX 11 and Theorem IX.12 However, an additional difficulty arises since one must specify in w hat sense the F ourier transform is the “ boundary v alu é” of the analytic function There is a wide range of such theorem s; we will discuss one which we will use in Section IX.8 in our study of quantum field theory Some of the others are briefly discussed in the Notes

D e fin itio n Let a e R", ja | = 1, and 6 e (0, n/2) Then

ra>9 = { ¿ e i r | ¿ - a > |^ | eos 0}

is called the cone about a of opening angle 0 The cone T * 9 = rBjB/2_8 is called the dual cone W hen no confusión arises, we will drop the subscripts and ju st refer to T and T*

The dual cone T* will either contain T (as in Figure IX 1) or be contained

in T N otice th at T* is the interior of the intersection of the half-spaces

[t] ( t] • £ > 0} for £ e T If T is the open forward light cone in R 4 (with the velocity of light equal to one), then T* is also the open forward light

cone Given an open cone C c R", we will denote by R" — iC the open región of those ( = <Aj — ir¡1, X2 — it}2, •••, A„ — irj„y e C" so that A =

<Al5 , A„> e R" and tj = <»?!, 6 C Rn — iC is called the tube with

base C.

We can now say w hat we m ean by “ boundary valué.”

D e fin itio n Let 5 e Sf”(Un) and suppose F(£) is a function analytic in

R" — iC for some cone C Suppose th at for each fixed t¡0 e C, F(A — irj0) is a

tem pered distribution (i.e has at m ost polynom inal growth in A) and that as

Suppose that T e £ f ’(Un) and that the support of T is contained in r o e for some a e R" and 0 e (0, n/2) If T is given by a function T(x), then we can directly extend T to the tube IR" — iT* 9 by

T(X — irf) = (2n)~"12 J a»' xT (x) dx

F or r j e F * g, the integral makes sense since

_ ^-|i)||x|cos(i),x) < e - \ x\d(n) where d(r¡) = |??| m inx£¿rí # cos(?7, x) = dist(?7, 5F* 9); see Figure IX 1 Since

that we may differentiate under the integral sign We conclude that

T(X — ir¡) is analytic in the tube R" — iT* 9 since it is infinitely differentiable

in the complex sense F urtherm ore, if q> e ^ (R " ), and rj0 e F * g, then

by the dom inated convergence theorem Thus, T is the boundary valué of

T(X — ir¡) in the tube R" — ¡T *9.

If T is not a function but is of the form P(D)G where G is a

polynom ially bounded continuous function with support in F a e , then we can define

In the sam e way as above, one can then show that T(X - ir¡) is analytic

in R" — ¿r*9 and that T is the boundary valué of T(X — ir¡) Thus, we wish to prove th at every tempered distribution T with support in a cone can be w ritten T = P(D)G for some partial differential operator P(D) and som e polynom ially bounded continuous function G with support in the cone

T o see that this is a strong statem ent the reader should recall that the analogous statem ent for com pact sets (rather than cones) is false F or

example, the delta function cannot be w ritten as P(D)G where G has support

at the origin

T h e o r e m IX.15 (B ros-E pstein-G laser lemma) Let T be a proper open

convex cone in R" and let T 6 ¿P'(W) have support in F Then there exists a polynom ially bounded continuous function G with support in F and

a partial differential operator P(D) so that T = P(D)G.

Proof Let {e¡}"= t be a basis for R" consisting of vectors in T Every vector

x e R" may be uniquely written x = t y¡ e¡ so that we may use {y(}"= i as

coordinates for R" Define

where 6 is the characteristic function of {x | x 6 R, x > 0} Then

F me C m_1(Rn) and Fm has support in T Furtherm ore, if Q{D) =

dn/ d y 1 ■ ■ dyn, then Q(D)m+1Fm = 5 as the reader can easily check We

will show that for m sufficiently large, the convolution T * Fm is a well- defined continuous function with support in T and that Q(D)m+l(T * Fm) =

T * Q(D)m+1F m = T * 6 = T.

If b e r, then F is contained in the interior of F — b so we may find a

C 00 function ip that equals one on F and has support in T — b (see Figure IX.2) Since T e there is an N so that

T(X — ir¡) — (2n) "l2P(i(X — ir¡)) e ~ i^ ~ i'l^ ’xG(x) dx

Fjy» ■ • •, y„) = • • • yW(yi) • • •

\T(cp)\ = \T(iPcp)\<C 1 Y l\x*D»(il,<p)IU

IctfáN

MSN U e r - 6

Thus, T has a unique extensión, continuous in ||| |||¡,, to those CN functions / for which (supp / ) n (I" — b) is compact.

Choose m = N + 1 and for y e R " define FN + 1 j,(x) = FN+1(y — x) Then

F n + j y is a CN function and (supp PN + y) n F — b is com pact (see Figure

IX.3) Further, the m ap y - * F N + Uy is |||- |||¡,-continuous and polynomially bounded in y (see Figure IX.2) Thus, G(y) = T * FN + l (y) = T (F N + Í y) is a

polynomially bounded continuous function and supp G e T since

(supp FN + Uy) n F = 0 i f y ^ T

Furtherm ore, if f e C q (Rn), then

( / * G)(y) = T ( ( f * FN + 1)~y) (IX.11)

These formulas are analogous to those in Theorem IX.4 Both may be proven by writing approxim ate Riem ann sums for the integráis on the right and then observing that the approxim ate Riem ann sums converge in ||| |||f,

If we now let j e(x) be an approxim ate identity, then by the proposition at

the end of Section IX 1,

Q(D'fl + 2G = lim Q{D)N + 2(je * G)

£ i o[by (IX 12)]

= lim T((Q(DYl + 2j c * FN + 1)~y)

£ i 0(by Theorem IX.4)

o f y '( R n) of a function T{) — ir¡) analytic in the tube IR" — ¡T * 9 Moreover,

T{X — ir¡) satisfies the estímate

| f (X - ir,)| < |P (2 - *>7)1(1 + [distfa, 3 r * , ) ] - w) (IX 13)

for a suitable polynom ial P and positive integer N.

Conversely, suppose F(X — ir,) is analytic in R" — ¡T* 9 an d satisfies the

weaker estim ates:

(i) F o r each r,0 e T* 9, there is a polynom ial P n<¡ in 2n variables so that

l ^ - ¿ ( ' ? o + '? ) ) | < | / 5J ¿ > r¡)\

for all l e R " and r, e F * e

(ii) There is an integer r > 0 so that for each r¡0 e r * 9> there is a polynom ial Qno so that

\F{X - itr,0)\ < M

for all X e IR" and te (0, 1]

Then there is a tem pered distribution T with support in the cone Fa 9 so that T is the boundary valué of F(X — ir¡) in the sense of Sf'{R") Moreover,

F can be recovered from T by the formula

Proof Suppose that T e S f'(W ) and supp T <= T a 9 By the B ros-E pstein-

Glaser lemma, there is a polynom ially bounded continuous function G with support in T a¡e so that T = P(D)G for some partial differential operator

P{D) From the discussion before the lemma, we already know that T is the

boundary valué of the function

T(X — ir¡) = (2n)~"/2P(i(X — ir¡)) f e~ iiX~‘n) ' xG(x) dx

Conversely, we suppose that F(X — it¡) is analytic in IR" — ¡T*\9 and that

the estimates (i) and (ii) hold The proof is in several steps First we show

that F(X — itr¡) has a tem pered distribution 7^ as boundary valué as t i 0 Then we show that the limit is independent of r¡ Finally we show that T has support in F a 9.

F or each fixed rj0 e T£\9, 0 < t < 1, F(X — itr¡0) is a well-defined distribu­

tion in ^ '(R " ) which we denote by 77 - Let i¡/ e £f(Un) and set

s c s u p

where k has been chosen large enough so that JR« (1 + |A|) k d i converges

Let p = r + 2 Then, by the fundam ental theorem of calculus,

where the Q} are suitable polynomials The estimates (IX.15) show that the limit of h(tx) exists as t x I 0 and that each term in the limit is less than or equal to a constant times an ^(R '^-sem in orm of ijj Thus F(2 — itr¡0) converges in ^ '( R " ) as t J, 0 to a tempered distribution which we will denote by 7o, , 0 Now suppose th at r¡x, r¡2 e T * g and that ¡¡/(x) is in

q ? (R n) T h e n ’

where we have used the fact that 11/(1) is entire and the estim ates on ij/(l)

in the Paley-W iener theorem to shift the hyperplane of integration in the

and therefore 7 o , = 7o,„(^) Since such i¡/ are dense in 5^(Rn), 7o,,, = 7o, , 2- Thus, the limit of T,.,0 as í J, 0 is independent of r¡0; we denote the limit by T.

We have shown that F(1 — irj) has a tem pered distribution f as a boundary valué It remains to prove that its inverse F ourier transform T

has support in r o 9 and to verify (IX.14) Let r¡x e T * 9 be given and suppose that <p 6 C “ (R") has com pact support in the open half-space {x | r¡x ■ x < 0} Then there is e > 0 so that x e supp <p implies r¡x ■ x < —e.

= | F(X — itrh)(p(X) dX

= J F(X — i(t + s)?;!)</>(/ — isrh) dX

(IX 16)

by Cauchy’s formula Thus, if we use the hypothesis (i) and choose N large

enough in the estím ate (IX 16), we find that for arbitrary s > 0,

and thus Tt ni((p) = 0 Therefore, the support of is contained in the half-space {x | • x > 0} for each t > 0 Since T, -> T as t j 0, we conclude that supp T cz {x | r¡! ■ x > 0} Since r „ 9 is the intersection of the closed sub- spaces {x | tjj • x > 0} where runs through r*9, we see that supp T

<= r a>9

Finally, suppose ¡¡/ e Cq (Rn) as above Then

f F(X — ir¡)^j(X) dk — f F (t — isr¡)\¡/(x — (s — l)ir¡) dx

= T ( e ~ i x{¡/)

= ( e " ’'-* T )(ír)

= ( e < ^ ) ( i P )

This proves (IX.4) and completes the proof of the theorem |

We will use this theorem in our study of axiom atic quantum field theory

in Section 9

IX.4 I f E stim ates

T here are a large num ber of Lp-estimates on F ourier transform s and

convolutions These estim ates are useful because they give conditions on p and q so that the F ourier transform or convolution by a given function is a bounded m ap from I í to 13 T he proofs of the estimates typically require

delicate use of í?-interpolation theorems In this section we State several Lp-interpolation theorem s and give examples to show how they may be used to derive estimates In the Appendix, we prove the first of these theorem s (Theorem IX 17) and use the idea of this proof to prove a variety

of other interpolation theorem s not used in this section

T he simplest íf-interpolation theorem is:

T h e o r e m IX 17 (The R iesz-T horin theorem ) Let <M, ¿t> and <N, v) be

m easure spaces with «r-finite measures Let 1 < p 0 , p u q0 , q i < oo and

suppose that T is a linear transform ation from I í o(M, dp) r> IS'{M, dp) to

U°(N, dv) n U '(N , dv) which satisfies ||T / ||9o < M 0| | / | | Po and ¡iT y ^, <

A^iII/IIp,- Then, for each f e H ° n l F ' , and each t e (0, 1), T f e 13' and

I|T /||9i< c,¡|/IIPi where p ' 1 = t p l1+ (1 - t ) p ¿ \ qt_1 = í ^ 1 + (1 - í ) 9 o \

and C, =

Notice that if the hypotheses of the theorem are satisfied, then T can

be extended by the B.L.T theorem to be a bounded map from LP'(M, dp)

to 13'(N, dv) The R iesz-Thorin theorem says essentially that the set of

\ < ’> so that T : H (M , dp) -> I3(N, dv) is bounded is a convex subset

of the plañe and that on that subset the logarithm of the norm of T is a

convex function The R iesz-Thorin theorem is a special case of the Stein interpolation theorem which is proven in the Appendix at the end of this section

We have already given one application of this theorem in Section 2 (the H ausdorff-Y oung theorem) Here is another:

E x a m p le 1 (Young’s theorem and inequality) W hen / and g are in

^ (R " ) we defined the convolution by the formula

Suppose p ~ 1 + q ~ 1 = 1 If f e LP(R") and g e I3(W), then the integral converges absolutely for all x by H older’s inequality Thus we can use (IX 17) to d e fin e /* g w h e n /e 12, g e 13 N ote that ||/* S f ||Q0< ||/ ||J s r ||4 Next, suppose that f g e L^R") Then

}} I f ( x - y ) g { y ) \ d x d y = | | / | | , ¡|sf||i

so by Fubini’s theorem the integral in (IX.17) exists a.e in x, and the

function f * g (defined a.e.) satisfies \ \ f * Sf||i < ||/ ||i ||g ||i - We can now use the R iesz-Thorin theorem to define the convolution on other U spaces Suppose that / 6 L^R") Then Tf (g) = / * g is a bounded operator from

L^R") to ¿ (R " ) (of norm less than or equal to | | / | | i ) and from L°°(R") to L°°(Rn) (of norm less than or equal to ||/ ||i ) Therefore, by the Riesz- Thorin theorem, 7}: LP(R") -> LP(R") and has norm less than or equal to

||XIIi- N ° w fix SfeLp(R") Then

Tg: Í?(R") -> L°°(R"), ||TJ| < ||sf||p