J. Fourier, 1808-9: "Regarding the researches of d’Alembert and Euler could one not add thatif they know this expansion, they made but a very imperfect use of it. They were both persuadedthat an arbitrary and discontinuous function could never be resolved in series of this kind, and itdoes not even seem that anyone had developed a constant in cosines of mulitiple arcs, the firstprobelm which I had to solve in the theory of heat."

FOURIER ANALYSIS

XU WANG

These notes (written from 2018-10-24) for TMA4170 are based on Dym–Mckean’s mono-graph [3], Stein–Shakarchi’s book [12] and Boggess–Narcowich’s book [1].

The purpose of the notes is to give a mathematical account of Fourier ideas on the circle andthe line. The emphasis is placed on the applications, which include

1). Eigenfunction expansion for the Dirac operator ´i ddx

;2). Wirtinger and Poincaré inequality;3). Heat equation on the circle;4). Weyl’s equidistribution theorem;5). Random walks;6). Poisson summation formula;7). Jacobi theta identities;8). Paley–Wiener theorem;9). Sampling theorem;10). Heisenberg uncertainty principle;11). Polynomial approximation;12). Gibbs’ phenomenon;13). Central limit theorem;......

CONTENTS

1. Fourier series 31.1. Main definition and some examples 31.2. The first question 51.3. Pointwise convergence 51.4. Proof of Theorem 1.1 by Chernoff [2] 61.5. Mean square convergence 81.6. Fourier series as an eigenfunction expansion 101.7. Some applications of Fourier series 121.8. Several dimensional Fourier series 18

Date: February 7, 2019.1

2 XU WANG

2. Fourier transform 222.1. Fourier transform on the Schwartz space 222.2. Classical Poisson summation formula 242.3. Fourier inversion formula and Plancherel identity 262.4. Fourier transform of tempered distributions 272.5. Fourier–Laplace transform and Paley–Wiener theorem 352.6. Sampling theorem 402.7. Heisenberg uncertainty principle 412.8. Central limit theorem 432.9. Fast Fourier transform 443. Wavelet analysis 444. Appendix 1: Definition of e, π and Euler’s formula 444.1. Definition of e 444.2. Definition of the exponential function 454.3. Definition of π and trigonometric functions 465. Appendix 2: Lebesgue integral 475.1. Lebesgue integral on r´π, πs 475.2. Lebesgue measure on Rn 486. Exercise sets 496.1. Exercise set 1: Fejér kernel and its applications 496.2. Exercise set 2: Gibbs’ Phenomenon 516.3. Exercise set 3: Isoperimetric inequality, Eigenfunction expansion and temperature

of the earth 526.4. Exercise set 4: Eigenfunctions of the Fourier transform 526.5. Exercise set 5: Poisson summation formula 537. TMA4170 (2019 Spring) Exercise 537.1. Week 2 537.2. Week 3 547.3. Week 4 557.4. Week 5-1 557.5. Week 5-2 577.6. Week 6 597.7. Week 7 597.8. Week 8 597.9. Week 9 607.10. Week 10-11 607.11. Week 12-15 60References 60

FOURIER ANALYSIS 3

1. FOURIER SERIES

1.1. Main definition and some examples. We shall follow Stein-Shakarchi’s book in this sec-tion. Fix a P R, L ą 0. Let f be a continuous function on ra, a` Ls.

Definition 1.1. The n-th Fourier coefficient of f is defined by

f̂pnq “ 1L

ż a`L

a

fpxqe´2πinx{L dx, n P Z.

The Fourier series of f is formally given by

fpxq „8ÿ

n“´8f̂pnqe2πinx{L.

Remark: At this point, we do not say anything about the convergence of the series. Thefollowings are examples from page 36–38 in [12]:

Example 1: Let fpxq “ x for x P r´π, πs. We have

f̂pnq “ 12π

ż π

´πxe´inx dx.

Sincepe´inxq1 “ p´inqe´inx,

if n ‰ 0 we have (by integration by parts)ż π

´πxe´inx dx “

ż π

´πx

ˆ

e´inx

´in

˙1

dx “ xˆ

e´inx

´in

˙

ˇ

ˇ

π

´π ´ż π

´π

ˆ

e´inx

´in

˙

dx “ 2πp´1qn

´in ,

which gives

f̂pnq “ p´1qn`1

in, n ‰ 0.

If n “ 0 thenf̂p0q “ 1

2π

ż π

´πx dx “ 0.

Thus

fpxq „ÿ

n‰0

p´1qn`1

ineinx.

Example 2: Fix s P CzZ and consider

fpxq “ πsinπs

eipπ´xqs, 0 ď x ď 2π.

By definition, we have

f̂pnq “ 12π

ż 2π

0

π

sin πseipπ´xqseinx dx “ e

iπs

2 sinπs

ż 2π

0

e´ipn`sqx dx,

4 XU WANG

integration by parts givesż 2π

0

e´ipn`sqx dx “ e´ipn`sqx

´ipn` sqˇ

ˇ

2π

0“ e

´2iπs ´ 1´ipn` sq .

Apply the Euler formula eix “ cosx` i sinx (see the appendix), we get

f̂pnq “ 1n` s

and

fpxq „ÿ

nPZ

einx

n` s.

Later we shall see that the right hand side is closely related to the Green function of ´i ddx` s on

R{2πZ, where ´i ddx

is called the one dimensional Dirac operator.

Example 3: The trigonometric polynomial defined for x P r´π, πs by

DNpxq :“Nÿ

n“´Neinx

is called the N -th Dirichlet kernel and is of fundamental importance in the theory (as we shallsee later). A closed formula for the Dirichlet kernel is

(1.1) DNpxq “sinpN ` 1

2qx

sin x2

, 0 ă |x| ă π; DNpxq “ 2N ` 1, x “ 0.

This can be seen by summing the geometric progressionsřNn“0 ω

n andř´1n“´N ω

n with ω “ eix.These sums are, respectively, equal to

1´ ωN`1

1´ ω , and1´ ω´N´1

1´ ω´1 ´ 1.

Their sum is then

ω´N ´ ωN`1

1´ ω “ω´N´

12 ´ ωN` 12

ω´12 ´ ω 12

“sinpN ` 1

2qx

sin x2

,

giving the desired result.

Example 4: The function Prpθq, called the Poisson kernel, is defined for θ P r´π, πs and0 ď r ă 1 by the absolutely and uniformly convergent series

Prpθq “ÿ

nPZr|n|einθ.

We know that the n-th Fourier coefficient of Pr is r|n| and

Prpθq “1´ r2

1´ 2r cos θ ` r2 ,

the proof is similar to the Dirichlet kernel case.

FOURIER ANALYSIS 5

1.2. The first question. Let us start form the following fact: all trigonometric polynomials areequal to their Fourier series. In fact, if f is a degree N trigonometric polynomial, i e

fpxq “Nÿ

n“´Ncne

inx

on ra, a` 2πs, then the lemma below implies

f̂pnq “ cn, |n| ď N ; f̂pnq “ 0, |n| ą N.

Lemma 1.1. If m ‰ n thenż a`2π

a

eimxeinx dx “ż a`2π

a

eipm´nqx dx “ 0;

if m “ n thenż a`2π

a

eimxeinx dx “ż a`2π

a

1 dx “ 2π.

We hope that a large class of functions have Fourier series expansion. A few reflections leadus to study the following question:

Let f be a 2π-periodic function on R. Find a natural condition on f such that all

f̂pnq :“ 12π

ż π

´πfpxqeinx dx

are well defined and the following series tfNpxquNě0 defined by

fNpxq :“Nÿ

n“´Nf̂pnqeinx

converges (in a certain sense) to fpxq.

1.3. Pointwise convergence. Denote by CkpS1q the space of all 2π-periodic Ck functions onR, we shall use the following extension of CkpS1q.

Definition 1.2. A 2π-periodic function f on R is said to be piecewise Ck pk “ 0, 1, ¨ ¨ ¨ q if thereexists

´π “ x0 ă x1 ă ¨ ¨ ¨ ă xm´1 ă xm “ π, m ě 1,such that for each 0 ă j ď m ´ 1, f |pxj ,xj`1q extends to a Ck function on a neighborhood ofrxj, xj`1s. We shall denote by PCkpS1q the space of all 2π-periodic piecewise Ck functions.

Remark: Let f P PC0pS1q. Then all its Fourier coefficients

f̂pnq “ 12π

m´1ÿ

j“0

ż xj`1

xj

fpxqeinx dx

are well defined. Lemma 1.1 suggests to define:

6 XU WANG

Definition 1.3. We callpf, gq “ 1

2π

ż π

´πfpxqgpxq dx,

the inner product on PC0pS1q and write ||f || “ pf, fq 12 .

Remark: It is clear that

f̂pnq “ pf, einxq, @ f P PC0pS1q,which gives

(1.2) fNpx0q “ pf,DNpx´ x0qq.We shall prove

Theorem 1.1. If f P PC1pS1q then

(1.3) limNÑ8

ˇ

ˇfNpxq ´fpx`q ` fpx´q

2

ˇ

ˇ “ 0.

Remark: For a general function f P PC0pS1q, we have

limNÑ8

ˇ

ˇ

f0pxq ` ¨ ¨ ¨ ` fN´1pxqN

´ fpx`q ` fpx´q2

ˇ

ˇ “ 0,

the proof is given in Exercise set 1 (see the end of the notes).

1.4. Proof of Theorem 1.1 by Chernoff [2].

1.4.1. Chernoff identity. Fix x0 P R, assume further that f is C1 near x0. Then

gpx´ x0q “fpxq ´ fpx0qeipx´x0q ´ 1

lies in PC0pS1q and gpyq is continuous near y “ 0 (try!). Notice that

fpxq “ peipx´x0q ´ 1qgpx´ x0q ` fpx0q.Change of variable y “ x´ x0 gives the following Chernoff identity (try!)

fNpx0q ´ fpx0q “ ĝp´N ´ 1q ´ ĝpNq.

1.4.2. Bessel inequality. Apply Lemma 1.1, we get

pgN , gq “ pgN , gNq.Thus we know that gN is orthogonal to g ´ gN , i.e.

pgN , g ´ gNq “ 0,which gives the following Bessel inequality

||g||2 “ ||g ´ gN ||2 ` ||gN ||2 ě ||gN ||2 “ÿ

|n|ďN

|ĝpnq|2.

FOURIER ANALYSIS 7

Remark: The Bessel inequality

||g||2 ěÿ

|n|ďN

|ĝpnq|2

is true for every g P PC0pS1q.

1.4.3. Riemann–Lebesgue lemma. The Bessel inequality gives the following Riemann–Lebesguelemma

|ĝpnq| Ñ 0, as |n| Ñ 8,

which proves (1.3) in case f is C1 at x.

Remark: The Riemann–Lebesgue lemma

|ĝpnq| Ñ 0, as |n| Ñ 8,

is true for every g P PC0pS1q.

1.4.4. Using Dirichlet kernel for the general case. Recall that

fNpx0q “1

2π

ż π

´πfpx0 ` xqDNpxq dx.

Since DN is even and fpx0 ` xq can be written as

fpx0 ` xq “ f1pxq ` f2pxq,

where

f1pxq :“fpx0 ` xq ` fpx0 ´ xq

2

is even and

f2pxq :“fpx0 ` xq ´ fpx0 ´ xq

2

is odd, we have

fNpx0q “ pf1, DNq.

Notice that (try!) f P PC1pS1q implies that f1 is C1 near x0, thus the previous argument gives

limNÑ8

|fNpx0q ´ f1p0q| “ 0.

Since f1p0q “ fpx0`q`fpx0´q2 , the proof of Theorem 1.1 is complete.

8 XU WANG

1.5. Mean square convergence.

Theorem 1.2. For every f P C1pS1q, we have

|fNpxq ´ fpxq|2 ď ||f 1||2 ¨2

N, @ x P R,

in particular ||fN ´ f ||2 ď ||f 1||2 ¨ 2N

Proof. Let N 1 ą N , we have

|fNpxq ´ fN 1pxq| “ |N 1ÿ

|n|“N`1

f̂pnqeinx| ďN 1ÿ

|n|“N`1

|f̂pnq|.

Notice that if n ‰ 0 then

f̂pnq “ 12π

ż π

´πfpxqe´inx dx “ 1

2π

ż π

´πfpxq

ˆ

e´inx

´in

˙1

dx.

Thus integration by parts gives

f̂pnq “ ´in¨ pf 1pnq.

Now we have

|fNpxq ´ fN 1pxq| ďN 1ÿ

|n|“N`1

|pf 1pnq| ¨ 1n.

Thus the Cauchy-Schwarz inequality gives

|fN ´ fN 1 |2 ď

¨

˝

N 1ÿ

|n|“N`1

|pf 1pnq|2˛

‚¨

¨

˝

N 1ÿ

|n|“N`1

1

n2

˛

‚.

SinceN 1ÿ

|n|“N`1

1

n2“ 2

N 1ÿ

n“N`1

1

n2ď 2

ż 8

N

dx

x2“ 2N

and Bessel’s inequality givesN 1ÿ

|n|“N`1

|pf 1pnq|2 ď ||f 1||2,

we have

|fNpxq ´ fN 1pxq|2 ď ||f 1||2 ¨2

N,

thus the theorem follows from Theorem 1.1 by letting N 1 Ñ 8. �

FOURIER ANALYSIS 9

Remark: Since teinxunPZ satisfies

peinx, einxq “ 1, peinx, eimxq “ 0, n ‰ m,

we know that teinxunPZ is an orthonormal family in C1pS1q. The above theorem says that everyelement in C1pS1q can be approximated by finite sums

ř

|n|ďN cneinx generated by teinxunPZ,

thus we know that

Theorem 1.3. teinxunPZ is an orthonormal basis in C1pS1q.

Another consequence of Theorem 1.2 is the following

Theorem 1.4 (Parseval’s identity). ||f ||2 “ř

nPZ |f̂pnq|2 for every f P C1pS1q.

Remark: A similar proof (use "Week 2" exercise 5) in fact implies that the Parseval’s identityis also true on PC1pS1q.

1.5.1. Completion of C1pS1q. Let us recall the following definitions

Definition 1.4. tgnunPN Ă C1pS1q is said to be a Cauchy sequence if for every j there exists Njsuch that

||gn ´ gm|| ă1

j, @ n,m ě Nj.

Two Cauchy sequences tgnunPN and thnunPN are said to be equivalent if for every j there existsNj such that

||gn ´ hn|| ă1

j, @ n ě Nj.

Denote by rtgnus the set of Cauchy sequences equivalent to tgnu, we call rtgnus the equivalentclass of tgnu.

One may check that (try!)

prtgnus, rthnusq :“ limnÑ8

pgn, hnq

is well defined.

Definition 1.5. We call the set of all equivalent classes of Cauchy sequences in C1pS1q with theabove inner product the completion of C1pS1q and denote it by L2pS1q or L2r´π, πs.

Remark: Notice thatf ÞÑ rtf, f, ¨ ¨ ¨ , us

defines an injective map from C1pS1q to L2pS1q. Thus we may look at C1pS1q as a subset inL2pS1q. We leave it as an exercise to check that C1pS1q is dense in L2pS1q. Thus Theorem 1.2implies

Theorem 1.5. L2pS1q is a separable complex Hilbert space with orthonormal basis teinxunPZ.

10 XU WANG

Remark: Anther way to look at L2r´π, πs is to use the Lebesgue integral theory (see Appen-dix 2), which gives the following isomorphism (the proof can be found in [13])

L2r´π, πs » tf PMr´π, πs :ż π

´π|f |2dx ă 8u{ „,

where Mr´π, πs denote the space of Lebesgue measurable complex valued functions on r´π, πsand

f „ g ô f “ g a.e. on r´π, πs.

1.6. Fourier series as an eigenfunction expansion. Let us look at the following Dirac operator

D :“ ´i ddx

: f ÞÑ ´if 1,

on C8pS1q. It satisfies the following property.

Lemma 1.2. pDf, gq “ pf,Dgq for every f, g P C8pS1q.

Proof. Recall that the inner product is defined by

pDf, gq “ 12π

ż π

´π´if 1pxqgpxq dx.

Thus the first formula follows directly from integration by partsż π

´πf 1pxqgpxq dx “ fpxqgpxq

ˇ

ˇ

π

´π ´ż π

´πfpxqg1pxq dx “ ´

ż π

´πfpxqg1pxq

where the second identity follows since f, g are 2π-periodic. �

Remark 1: In general, a linear operator T on C8pS1q is said to be self-adjoint ifpTf, gq “ pf, Tgq,

for every f, g P C8pS1q. The above lemma implies that D is self-adjoint. Moreover, we knowthat the square of D is the Laplacian operator 2 :“ ´ d2

dx2, that is the reason why we call D the

Dirac operator.

Remark 2: We know that all Hermitian matrices are diagonalizable with real eigenvalues.This fact is also true for our Dirac operator D, in fact we have

Dpeinxq “ npeinxq,we call einx the eigenfunction of D with eigenvalue n. Thus we may look at the Fourier seriesexpansion of f P C8pS1q as an eigenfunction expansion (compare it with the eigen-theory ofmatrices). Since teinxu generates C8pS1q, we know that D has no other eigenvalues.

Remark 3: Recall that if s P C is not an eigenvalue of a matrix M then M ´ s is invertible.Apply this fact to D, it is natural to ask whether D ` s is invertible in case s P CzZ. In fact, theFourier series expansion defines the inverse of D ` s directly as follows

pD ` sq´1f : y ÞÑÿ

f̂pnq einy

n` s, f P C8pS1q.

FOURIER ANALYSIS 11

Since f̂pnq “ ´inf̂ 1pnq and f is smooth we know that

ř

f̂pnq einyn`s also lies in C

8pS1q. Moreover,f̂pnq “ pf, einxq gives

`

pD ` sq´1f˘

pyq “ pf,Gq ,where G is defined as follows:

Definition 1.6. For every fixed s P CzZ, y P R, we call

Gpxq :“ÿ

nPZ

einpx´yq

n` s̄ ,

the y-Green function of D ` s.

Recall that in Example 2, we proved thatř

nPZeinx

n`s is the Fourier series expansion of

fpxq :“ πsin πs

eipπ´xqs, 0 ď x ď 2π.

Thus Theorem 1.1 gives

(1.4)ÿ

nPZ

einx

n` s “π

sin πseipπ´xqs, @ x P p0, 2πq,

and the following crucial identity in Eisenstein series (see page 5 in [4])

(1.5)ÿ

nPZ

1

n` s “fp0q ` fp2πq

2“ π cotπs,

i.e.ř

nPZeinx

n`s is equal to a 2π-periodic function g such that

gpxq “ πsin πs

eipπ´xqs, @ x P p0, 2πq

andgp0q “ π cot πs.

Now we know that G lies in PC8pS1q, a closed formula for G on r0, 2πs is the following:

Theorem 1.6. Assume that y P p0, 2πq and s P CzZ. We have Gpyq “ π cot πs̄,

Gpxq “ πsin πs̄

eipπ´x`yqs̄, @ x P py, 2πs

andGpxq “ π

sin πs̄e´ipπ`x´yqs̄, @ x P r0, yq.

Proof. The first two formulas following directly (1.4) and (1.5). The last formula follows fromÿ

nPZ

einx

n` s “ gpx` 2πq “π

sin πse´ipπ`xqs, @ x P p´2π, 0q,

(think of x as x´ y, s as s̄). �

12 XU WANG

Remark 1: Since G P PC8pS1q, by the mean square convergence theorem, we know that

||G´ÿ

|n|ďN

einpx´yq

n` s̄ || Ñ 0, N Ñ 8,

and||G||2 “

ÿ

nPZ

1

|n` s|2 .

The above theorem gives (by a direct computation)

||G||2 “ π sinhp2πIm sq| sinpπsq|2Im s “π sinhp2πIm sq

pcoshp2πIm sq ´ cosp2πRe sqqIm s.

Remark 2: Another way of looking at G is the following: G is the unique 2π-periodic distri-bution on R that solves

pD ` s̄qpGq “ 2πÿ

kPZδy`2πk,

where δξ is known as the Dirac distribution or Dirac’s delta function (we will come back to itlater).

Question: It is natural to ask the following questions:1. What will happen if s goes to zero and how to define the Green function for D itself ?

2. Can you develop similar theories for Dk (try the Laplacian D2 first)?

3. What is the relation between Green function of D and Green function of Dk.

1.7. Some applications of Fourier series.

1.7.1. Wirtinger and Poincaré inequality. We shall follow page 91 in [12]. The first version ofthe Wirtinger inequality (also called optimal one-dimensional Poincaré inequality) is

Theorem 1.7. Let f P C1pS1q with pf, 1q “ 0. Then||f || ď ||f 1||,

with equality if and only if fpxq “ f1pxq “ f̂p1qeix ` f̂p´1qe´ix.

Proof. By the proof of Theorem 1.2, we have

f̂pnq “ ´in¨ pf 1pnq.

Apply Bessel’s inequality to f 1, we have

||f 1||2 ěÿ

|n|ďN

|pf 1pnq|2 “ÿ

|n|ďN

n2|f̂pnq|2.

Since pf, 1q “ 0 implies that f̂p0q “ 0, thus the Parseval’s identity gives

||f ||2 “ limNÑ8

ÿ

0ă|n|‰N

|f̂pnq|2 ď ||f 1||2,

FOURIER ANALYSIS 13

with inequality if and only if f̂pnq “ 0 for all |n| ą 1, i.e. f “ f1. �

Remark: Notice that |f 1| “ |Df |, thus the above theorem gives||Df || ě ||f ||,

in case f P C8pS1qwith pf, 1q “ 0. The above identity is formally equivalent to that all non-zeroeigenvalues, say λ, of D satisfy

|λ| ě 1.Theorem 1.7 also implies

Proposition 1.1. Let f P C1pS1q with pf, 1q “ 0. Then for every g P C1pS1q, we have|pf, gq| ď ||f || ¨ ||g1||.

Proof. Notice thatpf, gq “ pf, g ´ ĝp0qq.

Thus the theorem follows from Cauchy–Schwarz inequality and Theorem 1.7. �

The second version of the Wirtinger inequality

Theorem 1.8. Let f be a C1 function in a neighborhood of r0, πs such that fp0q “ fpπq “ 0.Then

ż π

0

|fpxq|2 dx ďż π

0

|f 1pxq|2 dx,

with equality if and only if fpxq “ A sinx.

Proof. Check (try!) that f |r0,πs extends to an odd function (still denote it by f ) in C1pS1q, inparticular, pf, 1q “ 0, thus Theorem 1.7 applies. �

Remark: The above proof of the Wirtinger inequality also applies to the famous isoperimetricinequality (see page 103 in [12]). A natural high diemsional generalization of the convex versionof the isoperimetric inequality is the classical Brunn–Minkowski inequality (see [7]).

Theorem 1.9 (Brunn–Minkowski Theorem). Let A0, A1 be two compact convex sets in Rn withnon-empty interior. Then

|A0 ` A1|1n ě |A0|

1n ` |A1|

1n ,

where |A| denotes the Lebesgue measure (volume) of A andA0 ` A1 :“ tx` y P Rn : x P A0, y P A1u

is called the Minkowski sum of A0 and A1.

Remark: The above Brunn–Minkowski inequality is also true for every non-empty compactsets A0 and A1 (this general version is proved by Lazar Lyusternik in 1935), which can be seenas a generalization of the usual isoperimetric inequality.

14 XU WANG

1.7.2. Heat equation on the circle. We shall follow page 61–64 in [3].

The derivation of the heat equation is based on Newton’s law of cooling, which states that theflux of heat across a point x0 is proportional to the gradient of the temperature at x0 (this is anexperimental fact that is well verified for moderate temperature gradients, the full experimentalrelationship between flux and gradient is very complicated). This means that the amount of heatthat flows past x0 from left to right in a short time rt0, t0 ` δts is approximately

´c1 ¨ uxpx0, t0q ¨ δtwith a positive constant c1 depending upon the meterial (the minus sign is present because heatflows from the hotter place to the cooler). To proceed, the net amount of heat flowing out of asmall interval rx0 ´ δx, x0 ` δxs in a short time rt0, t0 ` δts is therefore

p´c1 ¨ uxpx0 ` δx, t0q ¨ δtq ´ p´c1 ¨ uxpx0 ´ δx, t0q ¨ δtq,or approximately so. This can be computed in a second way: It is, in fact, proportional to theproduct of the length of the interval and the (average) decrease of the temperature inside. Theconstant of proportionality is the "specific heat" of the conducting material. Therefore,

´c2utpx0, t0q2δx ¨ δt “ p´c1 ¨ uxpx0 ` δx, t0q ¨ δtq ´ p´c1 ¨ uxpx0 ´ δx, t0q ¨ δtq,or approximately so. Letting δx and δt go to zero, you find

utpx0, t0q “c1c2uxxpx0, t0q.

We can replace the time coordinate t by t “ cT , where c is another positive constant. If we setUpx, T q “ upx, cT q, then

UT px, T q “ cutpx, cT q “cc1c2uxxpx, cT q “

cc1c2Uxxpx, T q.

Choose c such that 2cc1 “ c2, it is enough to study the following standard heat equation

ut “1

2uxx.

In this section, we shall study the above heat equation on the circle R{Z (of length one, i.e. thetemperature is one-periodic with respect to x). Because u is supposed to be the temperature, it isnatural to conjecture that the whole solution is determined by the temperature

fpxq :“ upx, 0qat t “ 0. Since u is one-periodic with respect to x, we can formally write

upx, tq “ÿ

nPZcnptqe2inπx,

with

cnptq “ż 1

0

upx, tqe´2inπx dx.

It is enough to compute cnptq.

FOURIER ANALYSIS 15

Lemma 1.3. Each cnptq satisfies

cnp0q “ż 1

0

fpxqe´2inπx dx

andc1nptq “ ´2π2n2cnptq.

Proof. The first identity is just the initial condition. For the second identity, notice that

c1nptq “ż 1

0

utpx, tqe´2inπx dx “1

2

ż 1

0

uxxpx, tqe´2inπx dx.

Integration by parts givesż 1

0

uxxpx, tqe´2inπx dx “ż 1

0

upx, tqpe´2inπxqxx dx “ ´4π2n2ż 1

0

upx, tqe´2inπx dx.

Thus the lemma follows. �

Solving the above ODE gives

cnptq “ˆż 1

0

fpyqe´2inπy dy˙

e´2π2n2t,

thus

upx, tq “ÿ

ˆż 1

0

fpyqe´2inπy dy˙

e´2π2n2te2inπx “

ż 1

0

θpx´ y, 2πitq fpyq dy

where θ denotes the classical Riemann theta function defined by

(1.6) θpz, tq :“ÿ

nPZeπipn

2t`2nzq, z P C, t P H :“ tt P C : Im t ą 0u.

One may check that (try!) θ is holomorphic on CˆH.

Definition 1.7. We call θ the Jacobi theta function andhpx, y, tq :“ θpx´ y, 2πitq, x, y P R, t ą 0,

the heat kernel on the circle.

The heat kernel above solves the heat equation in the following sense

Theorem 1.10. For f P C2pR{Zq, put

upx, tq “ż 1

0

hpx, y, tq fpyq dy.

Then we have1) u is one-periodic with respect x and is smooth on Rˆ p0,8q;2) ut “ 12uxx on Rˆ p0,8q;3) limtÑ0` supxPR |upx, tq ´ fpxq| “ 0.

16 XU WANG

Proof. It is easy to check 1q and 2q since e´2π2n2t is a rapidly decreasing function of n. 3q followsfrom

|upx, tq ´ fpxq| ďÿ

nPZp1´ e´2π2n2tq|f̂pnq|,

and

|f̂pnq| “ |p2πinq´2 pf2pnq| ď p2πnq´2ż 1

0

|f2pxq| dx.

�

Remark: The above theorem is also true for f P C0pR{Zq, for the proof and the uniquenessof u, see page 64–65 in [3].

1.7.3. Weyl’s equidistribution theorem. We shall follow page 106–112 in [12], for related results,see page 54–56 in [3]. A basic postulate of statistical mechanics is the so called ergodic principleof Boltzman and Gibbs, which states that the time average of a mechanical quantity should bethe same as its phase average; see Ford and Uhlenbeck [6] (page 9–13) for a nice discussion ofsuch matters. A simple instance of this phenomenon can be seen in the following model due toWeyl in 1916.

As phase space, bring in the circle R{Z, pick a number 0 ă γ ă 1, and look at the rotationx ÞÑ x1 :“ x` γ

with addition modulo 1. The trajectory of the phase point x0 “ x is the arithmetic sequencex0 “ x, x1 “ x` γ, ¨ ¨ ¨ , xn “ x` nγ, ¨ ¨ ¨ ,

considered modulo 1. A mechanical quantity is a function f P PC0pR{Zq. Its time average is

limnÑ8

řn´1k“0 fpxkqn

,

assuming this limit to exist, while its phase everage is just the arithmetic meanż 1

0

fpxq dx.

Weyl proved that

Theorem 1.11. If γ is irrational then

limnÑ8

řn´1k“0 fpkγqn

“ż 1

0

fpxq dx,

for every f P C0pR{Zq.

Proof. Use Fejér’s theorem to approximate f uniformaly by trigonometric polynomials (see Ex-ercise set 1), it is enough to prove the theorem for f “ e2πimx. If m “ 0 then both sides are 1. Ifm ‰ 0 then

n´1ÿ

k“0fpkγq “

n´1ÿ

k“0ak, a :“ e2πimγ.

FOURIER ANALYSIS 17

Since γ is irrational, we know that a ‰ 1 andn´1ÿ

k“0ak “ 1´ a

n

1´ a

is bounded by 2|1´ a|´1. Thus

limnÑ8

řn´1k“0 fpkγqn

“ 0 “ż 1

0

fpxq dx.

The proof is complete. �

Remark: Fix two real numbers a ă b with b ´ a ă 1, consider the following one-periodicindicator function defined by

1ra,bs`Zpxq :“ 1 if x` n P ra, bs,

for some ineteger n and 1ra,bs`Zpxq :“ 0 otherwise. One may prove that the above theoremalso applies to 1ra,bs`Z: the idea is to approximate 1ra,bs`Z above and below by continuous func-tions f` and f´ so as to make

ş1

0pf` ´ f´q dx small and use the above theorem to f` and f´

respectively. Thus we get: if γ is irrational then

limnÑ8

#tk ă n : kγ P ra, bs ` Zun

“ b´ a,

and we say that tkγ`Zu is equidistributed in R{Z. A generalization of this fact is the followingWeyl’s criterion

Theorem 1.12. Let tξnu8n“0 be a sequence of real number. Then tξn ` Zu is equidistributed inR{Z if and only if for every nonzero integer k,

limNÑ8

řNn“0 e

2πikξn

N“ 0.

Proof. ñ: Since equidistributive property of tξn`Zu is equivalent to that for every one-periodicindicator function f ,

limnÑ8

řn´1k“0 fpξnqn

“ż 1

0

fpxq dx.

Approximating f P C0pS1q by one-periodic indicator function, we know that the above identityis also true for every f P C0pS1q. Now it is enough to apply it to fpxq “ e2πikx.

ð: Follows by a similar argument as in the proof of equidistributive property of tkγ`Zu. �

18 XU WANG

1.8. Several dimensional Fourier series. We shall follow page 81–85 in [3].

Definition 1.8. A function f on Rd is said to be Zd-invariant iffpx` kq “ fpxq, @ k P Zd.

We call a Zd-invariant function a function on the standard torus T d.

We shall denote by L2pT dq the completion of the space C8pT dq of smooth Zd-invariant func-tions with respect to the following inner product

pf, gq :“ż 1

0

¨ ¨ ¨ˆż 1

0

fpx1, ¨ ¨ ¨ , xdqgpx1, ¨ ¨ ¨ , xdq dx1˙

dx2 ¨ ¨ ¨ dxd, f, g P C8pT dq.

Since finite C-linear combinations of functions in t1|Qu, where Q denotes an arbitray n-cube,are dense in L2pT dq and

1|Qpxq “ 1|Q1px1q ¨ ¨ ¨ 1|Qdpxdq, Q :“ Q1 ˆ ¨ ¨ ¨ ˆQd,we know that

eZpxq :“ e2πiZ¨x, Z P Zd

defines an orthonormal basis of L2pT dq, which gives

1.8.1. Fourier series on a standard Torus.

Theorem 1.13. Every f P L2pT dq satisfies

limNÑ8

||f ´ÿ

ZPr´N,Nsdf̂pZq eZ || Ñ 0,

(later we shall write the above identity as f “ř

ZPZd f̂pZqeZ) where

f̂pZq :“ pf, eZq,denotes the Z-th Fourier coefficient of f .

1.8.2. Application to Random walks. Pólya discovered a very beautiful application of severaldimensional Fourier series to "random walks". Think of a particle moving on the d-dimensionallattice Zd according to the following rule. The particle starts at time 0 at the origin and moves attime n ě 1 by a unit step un to a neighborhood lattice point; for example, if d “ 3, the possiblesteps are

u “ p˘1, 0, 0q, p0,˘, 0q, p0, 0,˘1q, |u| “ 1.The position of the particle at time n ě 1 is the sum of the individual steps: sn “ u1 ` ¨ ¨ ¨ ` un.The step un is statistically independent of the preceding steps uj : j ă n and the possible stepsare equally likey at each stage. This means that

P pu1 “ û1, ¨ ¨ ¨ , un “ ûnq “ P pu1 “ û1q ˆ ¨ ¨ ¨ ˆ P pun “ ûnq “ p2dq´n

for any fixed unit steps û1, ¨ ¨ ¨ , ûn, in which P pEq means "the probability of the event E".The problem is to compute P psn “ Zq and to study the behavior of sn for nÑ 8.

FOURIER ANALYSIS 19

Pólya’s idea is to think of P psn “ Zq as the Fourier coefficient f̂pZq of a function f P L2pT nq:

fpxq “ÿ

P psn “ ZqeZpxq “ÿ

ZPZdP psn “ Zqe2πiZ¨x.

The sum is just the "expectation" or "mean value" of e2πisn¨x and is easily computed using theindependence of the individual steps. In fact, notice that

P psn “ Zq “ÿ

û1`¨¨¨`ûn“ZP pu1 “ û1q ˆ ¨ ¨ ¨ ˆ P pun “ ûnq “

ÿ

û1`¨¨¨`ûn“Zp2dq´n.

ThusP psn “ Zqe2πiZ¨x “

ÿ

û1`¨¨¨`ûn“Zp2dq´ne2πi pû1`¨¨¨`ûnq¨x,

which gives

fpxq “ÿ

ZPZd

ÿ

û1`¨¨¨`ûn“Zp2dq´ne2πi pû1`¨¨¨`ûnq¨x “

¨

˝p2dq´1ÿ

|u|“1

e2πiu¨x

˛

‚

n

.

Sinceÿ

|u|“1

e2πiu¨x “ 2 pcos 2πx1 ` ¨ ¨ ¨ ` cos 2πxdq ,

put

fdpxq :“cos 2πx1 ` ¨ ¨ ¨ ` cos 2πxd

d,

we getfpxq “ fdpxqn.

Thus Theorem 1.13 gives the following Pólya’s formula

P psn “ Zq “ f̂pZq “ pfnd , eZq.In particular,

P psn “ 0q “ż

r0,1sdfdpxqn dx1 ¨ ¨ ¨ dxd.

Since |fd| ď 1, the expected number of times the particle visits the origin can be expressed as8ÿ

n“0P psn “ 0q “ lim

εÑ1

8ÿ

n“0εnP psn “ 0q “ lim

εÑ1

ż

r0,1sd

8ÿ

n“0εnfdpxqn dx1 ¨ ¨ ¨ dxd.

Since

limεÑ1

ż

r0,1sd

8ÿ

n“0εnfdpxqn “ lim

εÑ1

ż

r0,1sdp1´ εfdq´1 “

ż

r0,1sdp1´ fdq´1,

we get8ÿ

n“0P psn “ 0q “

ż

r0,1sd

1

1´ cos 2πx1`¨¨¨`cos 2πxdd

dx1 ¨ ¨ ¨ dxd.

20 XU WANG

By the definition of ex in Appendix 1 and the Euler formula, we get

cosx “ 1´ x2

2` x

4

4!` ¨ ¨ ¨ ,

which implies thatx2

3ď 1´ cosx ď x

2

2,

when |x| is small enough. Thus (leave as an exercise)ř8n“0 P psn “ 0q ă 8 if and only if

ż

r0,εsd

1

x21 ` ¨ ¨ ¨ ` x2ddx1 ¨ ¨ ¨ dxd ă 8,

which is equivalent to d ě 3 (by using the polar coordinate). Pólya used this to prove a verystriking fact about the ultimate begavior of the walk.

Theorem 1.14. If d ě 3 thenP p lim

nÑ8|sn| “ 8q “ 1.

If d ď 2 thenP psn “ 0 infinitely oftenq “ 1.

Proof. If d ě 3 then we know that the expected number, say P , of times the particle visits theorigin is less than infinity. Denote by pn the probability of n actural number of visits. We knowthat

P “ p8 ˆ8`8ÿ

n“0npn ă 8, p8 `

8ÿ

n“0pn “ 1.

Thus we must have p8 “ 0 and the actual number of visits is less than infinity with probability1, and since the origin is not special in any way, the same must be true for every lattice point inZd. This means that for any R ă 8, the particle ultimately stops visiting the ball |Z| ă R, andthat is the same as to say

P p limnÑ8

|sn| “ 8q “ 1.

Now let us assume that d ď 2. At time n “ 1, the particle steps to one of the 2d nearestneighbors of the origin. The problem is to check that the probability p of ultimately returning tothe origin is 1. In fact, the probability of visiting the origin m or more times (including the visitat time n “ 0) is pm´1. Thus the probability of precisely m visits is

pm´1 ´ pm “ pm´1p1´ pq.If p ă 1 then p8 “ 0 and the expected number of visits would be

8ˆ 0`8ÿ

m“1mpm´1p1´ pq “ p1´ pq´1 ă 8,

contradicting the evaluation thatř8n“0 P psn “ 0q “ 8. The proof is finished; for additional

information on the subject, see Feller [5], pp 342–371. �

FOURIER ANALYSIS 21

1.8.3. Fourier series on a two dimensional Torus. Pick numbers a P R, b ą 0 and introduce the"non-standard" lattice L Ă R2 of all points of the form

ω “ jp1, 0q ` kpa, bq, pj, kq P Z2.

Definition 1.9. A function f on R2 is said to be L-periodic iffpx` p1, 0qq “ fpxq “ fpx` pa, bqq,

for every x P R2.

Definition 1.10. We call the set of all points ω1 P R2 such thatω1 ¨ ω P Z,

for every ω P L the dual lattice of L, and denote it by L1.

Exercise: Check that L1 is the lattice of points

ω1 “ jp1,´abq ` kp0, 1

bq, pj, kq P Z2.

L1 “ L if and only if a P Z and b “ 1.Exercise: Check that

eγpxq :“ e2πiγ¨x

is L-periodic if and only if γ P L1.Remark: One may look at the torus

TL :“ R2{Lby identifying opposite sides of the following "fundamental cell"

FL :“ ttp1, 0q ` spa, bq : 0 ď t, s ď 1u.Denote by C8pTLq the space of smooth L-periodic functions on R2. Let L2pTLq be the comple-tion of C8pTLq with respect to the following inner product

pf, gqL :“ż

FL

fpxqgpxq dx1 ¨ ¨ ¨ dxn, f, g P C8pTLq.

Then we have the following generalization of the standard torus Fourier series expansion.

Theorem 1.15. Every f P L2pTLq has the following orthogonal decomposition

f “ÿ

γPL1f̂pγqeγ,

wheref̂pγq :“ pf, eγqL.

Moreover, the following Plancherel identity holds

||f ||2L “ b ¨ÿ

γPL1|f̂pγq|2.

22 XU WANG

Proof. Think of f and eγ as functions of y1 “ x1 ´ abx2 and y2 “1bx2. This will bring you back

to the standard torus case. �

Exercise: Try to write down the details of the proof of the above theorem (notice that b is thearea of FL; if f is Z-periodic then

gpyq :“ fpy1 ` ay2, by2q,is Zn-invariant).

Exercise: Try to develop similar Fourier series theory for general high dimensional torus.

2. FOURIER TRANSFORM

Recall that if a smooth function on R is 2L-periodic then the following Fourier series expan-sion holds

fpxq “8ÿ

n“´8

ˆ

1

2L

ż L

´Lfpyqe´2πin

y2L dy

˙

e2πinx2L .

A small change in viewpoint leads at once to the Fourier integral: the idea is that the right handside is really a Riemann sum over a subdivision with spacing 1

2L, and with any luck, it should

approximate the integral

fpxq “ż 8

´8

ˆż 8

´8fpyqe´2πiyγ dy

˙

e2πixγ dγ

as L goes to8. This does not make too much sense for a periodic function f (the integral cannotconverge well), but it does suggest that something can be done to recover a nice function f fromits Fourier integral (or transform):

f̂pγq :“ż 8

´8fpyqe´2πiyγ dy

via the inverse Fourier integralq

f̂ “ż 8

´8f̂pγqe2πixγ dγ.

The purpose of the next two sections is to put this formal discussion on a solid mathematicalfoundation.

2.1. Fourier transform on the Schwartz space.

Definition 2.1. By the Schwartz space, say S, on R, we mean the space of all smooth functions,say f , on R such that

supxPR

|x|k|f plqpxq| ă 8,

for every non-negative integers k, l, where f plq denotes the l-th order derivative of f .

Examples: e´x2 P S but p1 ` x2q´1 does not belong to S; nor does e´|x|, but for a differentreason (try to verify this statement).

FOURIER ANALYSIS 23

Definition 2.2. The Fourier transform of a function f P S is defined by

f̂pγq :“ż 8

´8fpyqe´2πiyγ dy.

The inverse Fourier transform of a function g P S is defined by

ǧpxq “ż 8

´8gpγqe2πixγ dγ.

Remark: We have f̂pxq “ f̌p´xq.

The Fourier transform interchanges convolutions with pointwise products. Moreover, we have

Proposition 2.1. If f P S then1) {fpx` hq “ f̂pγqe2πihγ whenever h P R;2) {fpxqe´2πihx “ f̂pγ ` hq whenever h P R;3) {fpδxq “ δ´1f̂pδ´1γq whenever δ ą 0;4) zf 1pxq “ 2πiγf̂pγq;5) {´2πixfpxq “ d

dγf̂pγq.

Proof. We only prove 4) and leave the others as exercises (see page 136–137 in [12]). Integrationby parts gives

ż N

´Nf 1pxqe´2πixγ dx “ fpxqe´2πixγ

ˇ

ˇ

N

´N ` 2πiγż N

´Nfpxqe´2πixγ dx,

so letting N goes to infinity gives 4). �

Example: The above theorem can be used to prove the following crucial identity in Fouriertransform:

Proposition 2.2. The Gaussian function e´πx2 is the fixed point of the Fourier transform, moreprecisely, we have

ze´πx2 “ e´πγ2 .

Proof. Put fpxq “ ze´πx2 . Then we have

f 1pxq “ ´2πxfpxq

Thus 4) and 5) in the above Proposition give

d

dγf̂pγq “ {if 1pxq “ ´2πγf̂pγq.

24 XU WANG

Thus formally we have ddγ

ln f̂pγq “ 2πγ and f̂pγq “ ce´πγ2 (for a rigorous proof, one maycompute the derivative of f̂pγqe´πγ2 and show that it vanishes). Now it is enough to prove thatc “ 1. Notice that

c “ f̂p0q “ż

Re´πx

2

dx.

Thus c “ 1 follows from the Gaussian integral formula, see Week 4–Exercise 1. �

Remark 1: In general, it is also natural to study eigenvalue and eigenvectors of the Fouriertransform (see Exercise set 4, see also a recent result of Rodgers and Tao [16] on non-negativityof the De Bruijn–Newman constant).

Remark 2: The above proposition together with 3) imply that for every s ą 0,

(2.1) {e´πx2s “ s´ 12 e´πγ2{s,we shall use this identity to prove the Jacobi Theta Identity.

The fact that Fourier transform interchanges differentiation and multiplication can be used toprove (try!) the following result.

Theorem 2.1. If f P S then f̂ P S.

Definition 2.3. For every f, g P S, we call

f ‹ g : x ÞÑż 8

´8fpx´ yqgpyq dy,

the convolution of f and g.

Proposition 2.3. If f, g, h P S then1) f ‹ g P S;2) f ‹ g “ g ‹ f ;3) zf ‹ g “ f̂ ĝ;4) pf ‹ gq ‹ h “ f ‹ pg ‹ hq;5) pf ‹ gq1 “ f 1 ‹ g “ g1 ‹ f .

Proof. Exercise (see page 142–143 in [12]). �

2.2. Classical Poisson summation formula. We need a lemma to state the classical Poissonsummation formula.

Lemma 2.1. Let f be a continuous function on R such that

|fpxq| ď Cp1` x2q´1.Then

fT pxq :“ÿ

kPZfpx` kT q,

FOURIER ANALYSIS 25

defines a T -periodic continuous function on R such that

|fT pxq ´ fpxq| ď Cπ2

T 2, @ |x| ď T

2.

Proof. PutfNpxq :“

ÿ

|k|ďN

fpx` kT q.

Then we know that fN converges uniformly to fT . Thus fT is continuous and obviously fT isT -periodic. The final estimate follows from

|fT pxq ´ fpxq| ď2C

T 2

8ÿ

k“1

1

pk ´ 12q2“ C π

2

T 2@ |x| ď T

2.

�

Remark: If f P S then every n-th derivative f pnq of f fits the above lemma, thus fT is smoothand T -periodic. Apply the Fourier series expansion, we get

fT pxq “ÿ

nPZcne

2πin xT ,

wherecn “

1

T

ż

|y|ăT2

fT pyqe´2πinyT dy.

Notice thatż

|y|ăT2

fT pyqe´2πinyT dy “

ÿ

kPZ

ż

|y|ăT2

fpy ` kT qe´2πinyT dy “

ż

Rfpyqe´2πin

yT dy “ f̂

´n

T

¯

,

which gives

Theorem 2.2. If f P S thenfT pxq “

1

T

ÿ

nPZf̂´n

T

¯

e2πinxT .

In particular, we have the following Poisson summation formula (just take T “ 1, x “ 0)(2.2)

ÿ

nPZfpnq “

ÿ

nPZf̂pnq

Application to Jacobi theta identities: Apply the above theorem to

fpxq “ eπipx2t`2xzq, z, t P C, Im t ą 0,by (2.1), we get

f̂pλq “ p´itq´ 12 e´iπpλ´zq2

t .

Thus by the Poisson summation formula and the definition of theta function in (1.6), we get

(2.3) θpz, tq “ p´itq´ 12 e´ iπz2

t θ

ˆ

z

t,´1

t

˙

.

26 XU WANG

The readers can easily check the remaining theta identities

(2.4) θpz, t` 1q “ θˆ

z ` 12, t

˙

, θpz ` 1, tq “ θpz, tq, θpz, tq “ eπipt`2zqθpz ` t, tq.

Remark: Notice that

fT pxq “ÿ

kPZeπippx`kT q2t`2px`kT qzq :“ θpz, t;x, T q,

then the above theorem implies

(2.5) T ¨ θpz, t;x, T q “ÿ

nPZp´itq´ 12 e´iπ

p nT ´zq2t e2πin

xT “ p´itq´ 12 e´ iπz

2

t θ

ˆ

z

t` x,´1

t; 0,

1

T

˙

.

2.3. Fourier inversion formula and Plancherel identity. The Poisson summation formula im-plies the following result (for a direct Fourier series expansion proof, see page 89 in [3]).

Theorem 2.3. Every f in S satisfies the Fourier inversion formula

fpxq “ż 8

´8f̂pγqe2πixγ dγ

and the Plancherel identityż 8

´8|fpxq|2 dx “

ż 8

´8|f̂pγq|2 dγ.

Proof. By the definiton of Riemann integral, we haveż 8

´8f̂pγqe2πixγ dγ “ lim

TÑ8

1

T

ÿ

nPZf̂´n

T

¯

e2πinxT .

By Theorem 2.2, the above sum is equal to

fT pxq “ÿ

kPZfpx` kT q.

Thus we haveż 8

´8f̂pγqe2πixγ dγ ´ fpxq “ lim

TÑ8

ÿ

k‰0fpx` kT q,

but obviously the above limit is zero since f P S. Now let us prove the Plancherel identity.By Theorem 2.2 and the Plancherel identity for the Fourier series (recall that te2πinx{T u is anorthogonal basis of L2r´T {2, T {2s ), we have

ż

|x|ăT2

|fT pxq|2 dx “1

T

ÿ

nPZ

ˇ

ˇf̂´n

T

¯

ˇ

ˇ

2 Ñż 8

´8|f̂pγq|2 dγ, as T Ñ 8.

The left hand side goes toş8´8 |fpxq|

2 dx as T goes to infinity, thus the theorem follows. �

FOURIER ANALYSIS 27

Denote by

pf, gq :“ż 8

´8fpxqgpxq dx,

the inner product space structure on S. Put

||f || :“ pf, fq 12 .

Definition 2.4. We call the set of all equivalent classes of Cauchy sequences in S with the aboveinner product the completion of S and denote it by L2pRq.

Remark: Anther way to look at L2pRq is to use the Lebesgue integral theory (see Appendix2), which gives the following isomorphism

L2pRq » tf PMpRq :ż 8

´8|f |2dx ă 8u{ „,

where MpRq denote the space of Lebesgue measurable complex valued functions on R andf „ g ô f “ g a.e. on R.

The Plancherel identity implies that both the Fourier transform f̂ and the Fourier inversionpf̂q_ extend to L2pRq on which the Fourier inversion formula and the Plancherel identity stillhold. See Exercise set 4 for a canonical basis of L2pRq using eigenfunctions of the Fouriertransform.

2.4. Fourier transform of tempered distributions. In applications (e g elementary solution orGreen’s function of a partial differential operator, see Theorem 7.1.20 in [8] and the Week 5-2exercise 5), it is crucial to extend Fourier transforms to a larger class of functions. A natural wayto do it is to use the notion of distribution introduced by Schwartz.

Test functions: Denote by DR the space of smooth functions, say f , on R such thatfpxq “ 0, if |x| ě R.

PutD “ YRą0DR.

We call D the space of test functions. It is clear that D is a subspace of S.Exercise: D is not empty. Check that the classical cut-off function (see Wikipedia for "Molli-

fier")

(2.6) χpxq :“ ce1

|x|2´1 , |x| ă 1; χpxq :“ 0, |x| ě 1,is smooth on R, where c is choosing such that

ş

R χdx “ 1. For ε ą 0, putχεpxq :“ ε´1χpε´1xq,

then χε P Dε. Moreover, for every δ ą ε,

χε,δpxq :“ż

|y|ďδχεpx´ yq dy

28 XU WANG

lies in Dε`δ and χε,δpxq “ 1 if |x| ď δ ´ ε.

Definition 2.5. A C-linear mapT : D Ñ C

is said to be a distribution on R if for every R ą 0 there exists a positive constant CpRq and apositive integer NpRq such that

|T pfq| ď CpRq sup|x|ăR, 0ďnďNpRq

|f pnqpxq|, @ f P DR.

A distribution T on R is said to be tempered if it extends to a C-linear map, still denote it by T ,T : S Ñ C,

such that there exists a positive constant C and a positive integer N with

|T pfq| ď C||f ||N , ||f ||N :“ supxPR, 0ďk,lďN

|x|k|f plqpxq|, @ f P S.

We shall denote by D1 the space of all distrubutions on R and by S 1 the space of all tempereddistributions on R.

Remark: Notice that (try!)

limnÑ8

||f ´ χ1,nf ||N Ñ 0 “ 0, @ f P S

implies that every tempered distribution is uniquely determined by its restriction on D.Examples of distribution:

Piecewise continuous functions: If f is a piecewise continuous function then

Tf : g ÞÑż

Rfpxqgpxq dx,

defines a distribution (sometimes we identity f with Tf ). Check that Tex is not tempered.

Dirac’s delta function: Fix ξ P R, the Delta funcionδξ : f ÞÑ fpξq,

defines a distribution on R, moreover δξ P S 1

L2-functions: One may look at L2pRq as a subspace of S 1: in fact, for every f “ rtfnus,

Tf : g ÞÑ limnÑ8

ż

Rfnpxqgpxq dx, g P S,

defines a tempered distribution (we will identify f with Tf ) sinceˇ

ˇ

ż

Rfnpxqgpxq dx

ˇ

ˇ ďż

R|fnpxq|p1` x2q´

12 dx ¨ sup

xPR|p1` x2q 12 |gpxq|

andż

R|fnpxq|p1` x2q´

12 dx ď ||fn||

ˆż

Rp1` x2q´1 dx

˙12

“ π 12 ¨ ||fn||

FOURIER ANALYSIS 29

give

|Tf pgq| ď p2πq12 ¨ ||f || sup

xPR, 0ďkď1|x|k|gpxq|.

Cauchy principal values: The Cauchy principal value

p.v.

ˆ

1

x

˙

: f ÞÑ limεÑ0

ż

|x|ąε

fpxqx

dx

defines a tempered distribution. In fact, we haveż

|x|ąε

fpxqx

dx “ż 8

ε

fpxq ´ fp´xqx

dx.

Sinceˇ

ˇ

fpxq ´ fp´xqx

ˇ

ˇ “ˇ

ˇ

1

x

ż x

´xf 1ptq dt

ˇ

ˇ ď 2 sup|t|ďx

|f 1ptq|, x ą 0,

we getˇ

ˇ

ż 1

ε

fpxq ´ fp´xqx

dxˇ

ˇ ď 2 supxPR

|f 1pxq|,

andˇ

ˇ

ż 8

1

fpxq ´ fp´xqx

dxˇ

ˇ ď 2ˆ

supxPR

|xfpxq|˙ż 8

1

1

x2dx “ 2 sup

xPR|xfpxq|.

Thusˇ

ˇ

“

p.v.

ˆ

1

x

˙

‰

pfqˇ

ˇ ď 2 supxPR

|f 1pxq| ` 2 supxPR

|xfpxq|,

which implies that p.v.`

1x

˘

defines a tempered distribution.

Derivative of a distribution. Let T P D1. The derivatives of T are always well defined

T pkq : f ÞÑ T pp´1qkf pkqq.

Notice that T P S 1 implies that T pkq P S 1. If g is a smooth function then T pkqg “ Tgpkq .

Multiplication by smooth functions: Let f P D, T P D1. Then

fT pgq :“ T pfgq,

defines fT P D1. It is easy to check that

xT P S 1, fT P S 1,

if f P S, T P S 1.

30 XU WANG

2.4.1. Fourier transform of a tempered distribution. The motivation for extending Fourier trans-form to tempered distributions comes from the following identity (which is sometimes called themultiplication formula)

Theorem 2.4. If f, g P S thenż 8

´8fpxqĝpxq dx “

ż 8

´8f̂pyqgpyq dy.

Proof. Put F px, yq “ fpxqgpyqe´2πixy then the theorem follows fromż 8

´8

ˆż 8

´8F px, yq dy

˙

dx “ż 8

´8

ˆż 8

´8F px, yq dx

˙

dy,

the details are left to the readers. �

Definition 2.6 (Fourier transform on S 1). Let T P S 1, put

T̂ : f ÞÑ T pf̂q, Ť : f ÞÑ T pf̌q,

we call T̂ the Fourier transform of T and Ť the inverse Fourier transform of T .

Remark: The following lemma implies that T̂ , Ť P S 1 if T P S 1.

Lemma 2.2. For every f P S, we have

||f̂ ||N ď CpNq||f ||N`2

Proof. Follows from

|xkf̂ plqpxq| “ | {pylfqpkqpxq| ďż

R|pylfqpkq| dy ď π sup

yPRp1` y2q|pylfqpkq|.

�

One may define f ‹ T (f P S, T P S 1) as follows

pf ‹ T qphq :“ T pf´ ‹ hq, h P S,

where f´pxq :“ fp´xq. Then the following basic properties of the Fourier transform can benaturally generalized to tempered distributions.

Theorem 2.5. For every f, g P S, T P S 1, we have1) zf ‹ T “ f̂ T̂ ;2) yT pkq “ p2πixqk T̂ ;3) {´2πixT “ pT̂ qp1q;4) ˇ̂T “ T .

FOURIER ANALYSIS 31

Example: Fourier transform of the Delta function δξ:

pδξpfq “ δξpf̂q “ f̂pξq “ Te´2πixξpfq.

Thuspδξ “ Te´2πixξ

and the Fourier inversion formula gives

{Te2πixξ “ δξ.

Sometimes we just write pδξ “ e´2πixξ and ze2πixξ “ δξ.

2.4.2. Poisson summation formula and periodic distributions. We shall follow page 177–181 in[8]. By Lemma 2.1, we know that

u :“ÿ

kPZδk,

defines a tempered distribution. From the definition, we know that the Poisson summation for-mula is equivalent to

Theorem 2.6. The Fourier transform of u is equal to itself.

Definition 2.7. A distribution T P D1 is said to be 1-periodic if

T pfq “ T pfnq, @ f P D, n P Z,

where fnpxq :“ fpx` nq.

It is clear that u is 1-periodic. In general, let T P D1 be 1-periodic, put

φnpxq “χpx` nq

ř

kPZ χpx` kq,

where χ is defined in (2.6), then φ0 P D1 andř

nPZ φn “ 1 gives

T pfq “ÿ

nPZT pfφnq “

ÿ

nPZT pfnφ0q, @ f P D.

The above formula also implies that every 1-periodic distribution is tempered. Together with thePoisson summation formula

ÿ

nPZf̂px` nq “

ÿ

nPZfpnqe´2πinx

it gives

T pf̂q “ÿ

nPZT pe´2πinxφ0qfpnq.

Thus we get

32 XU WANG

Theorem 2.7. If T P D1 is 1-periodic then

T̂ “ÿ

nPZTR{Zpe´2πinxqδn,

where TR{Zpe´2πinxq :“ T pe´2πinxφ0q does not depend on φ0.

Remark: In case T “ Tg for a 1-periodic continuous function g, we have

T pe´2πinxφ0q “ż 1

0

gpxqe´2πinx dx “ ĝpnq

Moreover, by the Fourier inversion formula, the above theorem givesż

Rfpxqgpxq dx “ T pfq “ T̂ pxf´q “

ÿ

nPZĝpnq

ż

Rfpxqe2πinx dx,

which can be seen as a generalization of the Fourier series expansion gpxq „ř

nPZ ĝpnqe2πinx.

2.4.3. Elementary solution of D ` s. Recall that the Dirac operator D is defined by ´i ddx

.

Definition 2.8. Fix ξ P R, s P Cz2πZ. If a 1-periodic distribution T satisfies

pD ` sqT “ÿ

kPZδξ`k,

then we call T an elementary solution of D ` s on R{Z.

Assume that T is an elementary solution. Apply the Fourier transform to the equation, Theo-rem 2.7 gives

p2πx` sqT̂ “ÿ

nPZe´2πinξδn.

Since s P Cz2πZ, we get

T̂ “ÿ

nPZ

e´2πinξ

2πn` sδn,

which gives uniqueness of T . Moreover, since qδn “ e2πinx, one may guess that T is given by thefollowing function

fpxq “ÿ

nPZ

e2πinpx´ξq

2πn` s .

Recall that we have proved that

π

sin πseipπ´xqs “

ÿ

nPZ

einx

n` s, if x P p0, 2πq, π cotπs “ÿ

nPZ

1

n` s.

in Theorem 1.6, from which we get

FOURIER ANALYSIS 33

Theorem 2.8. The elementary solution of D ` s on R{Z is unique and can be written as Tf ,where f is a piecewise smooth 1-period function. When ξ P p0, 1q, we have (see the remarkbelow)

fpxq “ ieispξ´xqˆ

Hpx´ ξq ` 1eis ´ 1

˙

,

when x P r0, 1sztξu and

fpξq “ iˆ

1

2` 1eis ´ 1

˙

“ 12

cotps{2q.

Remark: Assume that ξ P p0, 1q. Another way to find the elementary solution is to find apiecewise smooth function on r0, 1s such that

´if 1pxq ` sfpxq “ δξ, fp0q “ fp1q,where the first equation is defined in the sense of distribution. Let us rewrite the equation as

pfeisxq1 “ ieisξδξ.It is easy (try!) to check that the following Heaviside function

Hpxq “ 0, x ă 0, Hpxq “ 1, x ą 1, Hp0q “ 12

satisfies H 1 “ δ0. Thus we havefeisx “ ieisξHpx´ ξq ` C.

Now fp0q “ fp1q gives

C “ ieisξ

eis ´ 1 .

Thus we get

fpxq “ ieispξ´xqˆ

Hpx´ ξq ` 1eis ´ 1

˙

,

when x P r0, 1sztξu and

fpξq “ iˆ

1

2` 1eis ´ 1

˙

“ 12

cotps{2q.

2.4.4. Wave and heat equations associated to D.Definition 2.9. Let D be a selfadjoint operator, we call B{Bt`D the heat operator associated toD and B2{Bt2 `D the wave operator associated to D.

Remark 1: In case D is the Dirac operaror ´i ddx

, then we can have

B{Bt`D “ B{Bt´ iB{Bx.Consider the complex coordinate

z :“ x` it, z̄ :“ x´ it

34 XU WANG

then we haveB{Bz̄ “ 1

2pB{Bx` iB{Btq “ i

2pB{Bt`Dq .

Thus up to a constant, the heat operator associated toD is equal to the Cauchy–Riemann operatorB{Bz̄ whose kernel gives the holomorphic functions.

Remark 2: Still let D be the Dirac operaror ´i ddx

, then the wave operator associated to D is

B2{Bt2 ´ iB{Bx,this operaor can be seen as a time-dependent Schrödinger operator with trivial potential energy,see [15] for a nice introduction of the Schrödinger equation.

Remark 3: In case D is the Laplacian ´d2{dx2, i e square of the Dirac operator, thenB{Bt´ B2{Bx2

is just the classical heat operator and

B2{Bt2 ´ B2{Bx2

is the classical wave operator .

2.4.5. Fourier transform of the Heaviside function. Recall that

2πix{H ´ C “ xH 1 “ δ̂0 “ 1,thus one might guess that

{H ´ C “ 12πi

p.v.

ˆ

1

x

˙

and C should be 12

since the right hand side is odd.

Theorem 2.9. {H ´ 12“ 1

2πip.v.

`

1x

˘

.

Proof. By definition, for every f P S, we have{

H ´ 12pfq “ 1

2

ˆż 8

0

f̂pxq dx´ż 0

´8f̂pxq dx

˙

“ 12

limkÑ8

ż k

1k

f̂pxq ´ f̂p´xq dx.

Definition of f̂ and the Euler formula give

f̂pxq ´ f̂p´xq2

“ ´iż

Rfpyq sinp2πxyq dy.

Thus{

H ´ 12pfq “ 1´2πi limkÑ8

ż

yPRfpyqcosp2πkyq ´ cosp2πk

´1yqy

dy,

Since y´1 cos ay is odd, we haveż

yPRfpyqcosp2πkyq ´ cosp2πk

´1yqy

dy “ż 8

0

pfpyq ´ fp´yqqcosp2πkyq ´ cosp2πk´1yq

ydy.

FOURIER ANALYSIS 35

Since f P S, we haveˇ

ˇ

ż 8

N

pfpyq ´ fp´yqqcosp2πkyq ´ cosp2πk´1yq

ydyˇ

ˇ ď N´1,

when N is large enough. Moreover, the Riemann-Lebesgue lemma implies that

limkÑ8

ż N

0

fpyq ´ fp´yqy

cosp2πkyq dy “ 0

since fpyq´fp´yqy

is continuous on r0, N s. Thus

{

H ´ 12pfq “ 1

2πilimNÑ8

limkÑ8

ż N

0

fpyq ´ fp´yqy

cosp2πk´1yq dy “ 12πi

ż 8

0

fpyq ´ fp´yqy

dy.

Thus the theorem follows. �

2.5. Fourier–Laplace transform and Paley–Wiener theorem. The aim is study the Fourier–Laplace transform and prove the Paley–Wiener theorem with the help of Cauchy integral formulaand the maximum principle in complex analysis. We shall follow Chapter 1 of Hörmander’s book[9] in this section. Our starting point is the following basic observation: if g P DR then

ĝpzq “ż

Rgpyqe´2πiyz dy “

ż R

´Rgpyqe´2πiyz dy,

defines a holomorphic function on z P C, i.e. ĝ is an entire function.

Definition 2.10. We call ĝpzq, z P C the Fourier–Laplace transform of g P DR.

Let us writez “ a` ib, a, b P R

then

ĝpa` ibq “ż R

´Rgpyqe´2πiaye2πby dy

gives

|ĝpa` ibq| ď e2πR|b|ż R

´R|gpyq| dy.

Moreover, for every natural number k, integration by parts gives

ygpkq “ p2πizqkĝpzq.

Thus

p2πqk|zkĝpzq| ď e2πR|b|ż R

´R|gpkqpyq| dy.

The smooth version of the Paley–Wiener theorem can be seen as an inverse statement of theabove estimate.

36 XU WANG

Theorem 2.10 (Paley–Wiener–Schwartz). Let U be an entire function such that

(2.7) p1` |z|qk |Upzq| ď Ck e2πR |Im z|,for every non-negative integer k. Then there exists u P DR whose Fourier–Laplace transform isU .

Proof. By the Fourier inversion formula, it suffices to define

(2.8) upxq “ż

RUpyqe2πixy dy

and check that u P DR. By (2.7), we know that U |R P S, thus u P S. Now it is enough to checkthat upxq “ 0 if |x| ą R. The key of the proof is to use the Cauchy integral formula: notice that(2.7) permits us to shift the integration in (2.8) into the complex domains, which gives

upxq “ż

RUpy ` ibqe2πixpy`ibq dy,

for every b P R. Estimating the integral by means of (2.7) with k “ 2, we obtain

|upxq| ď C2e´2πxb`2πRbż

yPRp1` |y|q´2 dy,

and the integral is convergent. If we choose b “ tx and let tÑ 8, it now follows that upxq “ 0if |x| ą R. �

In the next section, we shall study a distribution version of the above theorem.

2.5.1. Fourier–Laplace transform of distributions with bounded support.

Definition 2.11. The support of a complex function, f , on R is defined as

Supp f :“ tx P R : fpxq ‰ 0u.Let K be a bounded subset in R. We shall denote by EpKq the space of smooth functions on Rwith support in K.

Definition 2.12. Let T be a distribution on R. Fix x P R, we say that T is zero at x if there existsε ą 0 such that

T pfq “ 0, @ f P Eprx´ ε, x` εsq.The support SuppT of T is defined as the set of points where T is not zero. We say that T hascompact support if SuppT is bounded. Let K be a bounded subset in R. We shall denote byE 1pKq the space of distributions with support in K.

Example: Let g be a continuous function on R with support in r´R,Rs. Then for eachk ě 1, the k-th order derivative, T pkqg , of Tg lies in E 1pr´R,Rsq. In fact, every distributionon E 1pr´R,Rsq can be represented in this way (see [9] for related results). In particular, everydistribution with bounded support is tempered.

FOURIER ANALYSIS 37

Remark: One may prove that (try! or see page 11 in [9]) a distribution T lies in E 1pKq if andonly if there exist constant C and k such that

|T pfq| ď Cÿ

xPK, 0ďnďk|f pnqpxq|, @ f P C8pRq.

In particular, for every z P C, y ÞÑ e´2πiyz is smooth, thus the following functionT̂ : z ÞÑ T pe´2πiyzq

is well defined in case the distribution T has a bounded support.

Definition 2.13. We call T̂ the Fourier–Laplace transform of T .

Remark: By the fact in the Example above, we know that (try!) if T P E 1pr´R,Rsq then T̂is a holomorphic function on C such that

p1` |z|q´k |T̂ pzq| ď Ck e2πR |Im z|,for some integer k and constant Ck. The inverse statement is the following distribution versionof Theorem 2.10.

Theorem 2.11 (Distribution version of the Paley–Wiener theorem). Let U be an holomorphicfunction on C such that(2.9) p1` |z|q´k |Upzq| ď Ck e2πR |Im z|,for some integer k. Then there exists a unique T P E 1pr´R,Rsq such that T̂ “ U .

Proof. First note that U P S 1, thus we can write U “ T̂ for some T P S 1. Then {χε ‹ T “ χ̂εT̂satisfies Theorem 2.10 withR replaced byR`ε. Thus each Supp pχε‹T q lies in r´R´ε, R`εsand when ε goes to zero this implies that T P E 1pr´R,Rsq. �

The above theorem gives directly the following weakL2-version of the Paley–Wiener theorem.

Theorem 2.12. A holomorphic function U on C is the Fourier–Laplace transform of an L2 func-tion on r´R,Rs if and only if

p1` |z|q´k |Upzq| ď Ck e2πR |Im z|,for some integer k and the L2-norm of U on the real line is finite.

Remark: If we look at L2r´R,Rs as the completion of D with respect to the following norm

||f ||2 :“ż R

´R|fpxq|2 dx, f P D,

then Fourier–Laplace transform of f “ rtfnus P L2r´R,Rs can be defined as

f̂pzq :“ limnÑ8

ż R

´Rfnpxqe´2πixz dx.

The classical L2-version of the Paley–Wiener theorem will be proved in the next section.

38 XU WANG

2.5.2. Classical L2-version of the Paley–Wiener theorem. We shall use a version of Phragmén–Lindelöf Theorem in page 108–109 in [14].

Definition 2.14. We say that a A holomorphic function F on C is of exponential type T ą 0 iffor every ε ą 0 there exists a constant Aε such that

|F pzq| ď AεepT`εq|z|,

on C.

Theorem 2.13 (Paley–Wiener theorem). A holomorphic function U on C is the Fourier–Laplacetransform of an L2 function on r´R,Rs if and only if U is of exponential type 2πR and

ż

R|Upxq|2 dx ă 8.

We will prove the following stronger version the Paley–Wiener theorem.

Theorem 2.14 (Strong Paley–Wiener theorem). Let U be a holomorphic function on C whoserestriction to the real line is L2. Then the followings are equivalent:

1) U is of exponential type 2πR;2) For all y P R, we have

ż

R|Upx` iyq|2 dx ď e4πR|y|

ż

R|Upxq|2 dx;

3) For all ε ą 0, we have

|Upzq|2 ď 2e4πRp|Im z|`εq

πε

ż

R|Upxq|2 dx;

4) U is the Fourier–Laplace transform of an L2 function on r´R,Rs;5) p1` |z|q´k |Upzq| ď Ck e2πR |Im z| for some integer k.

Proof. It is clear that 5q implies 1q. The fact that 2q implies 3q follows from the sub-meaninequality (try to prove it using taylor expansion at z0)

|Upz0q|2 ď1

πε2

ż

|z´z0|ăε|Upzq|2 dxdy.

For 3q implies 4q, by the weak L2-version of of the Paley–Wiener theorem, we know that U isthe Fourier Laplace transform of an L2 function, say u, on r´R ´ ε, R ` εs for every ε ą 0.Since u is uniquely determined by U , we know that the support of u lies in r´R,Rs. Now theweak L2-version of the Paley–Wiener theorem gives that 4q implies 5q. Now it suffices to showthat 1q implies 2q (will be proved at the end of this section). The key is the following variant ofmaximum principle (so called the Phragmén–Lindelöf Theorem). �

FOURIER ANALYSIS 39

Theorem 2.15 (Phragmén–Lindelöf Theorem). Assume that f is holomorphic and |fpzq| ďAe|z|

βin a sector D of (angular) opening less than π{α. If f is also continuous in the closed

section D̄ and 0 ď β ă α thensupzPD̄

|fpzq| “ supzPBD

|fpxq|

Proof. Assume that supzPBD |fpxq| “M . By a rotation, one may assume thatD “ treiθ : r ą 0, |θ| ă ψu,

where ψ ă π2α

. Pick β ă γ ă α and define

F pzq “ fpzqe´εzγ ,for every ε ą 0. It follows that |F pzq| ď |fpzq| ď M on the two bounding rays. Moreover, onthe arc z “ Reiθ|, |θ| ď π

2α,

|F pzq| ď AeRβ´εRγ cospγπ{2αq,which tends to zero asR goes to infinity. Thus |F pzq| ď 1

Nfor everyN provedR is large enough.

The maximum principle, thus, implies that

|F pzq| ď limNÑ8

maxt1{N,Mu “M

on D̄, which gives that|fpzq| ďMeεrγ cospγθq

for all z “ reiθ P S̄, Letting εÑ 0 we obtain our result. �

Lemma 2.3. Suppose F is of exponential type T and |F pxq| ď 1 for x real then |F px ` iyq| ďeT |y| for all complex numbers z “ x` iy.

Proof. For ε ą 0 setFεpzq “ F pzqeipT`εqz.

Since F is of exponential type T

|Fεpiyq| “ |F piyq|e´pT`εqy ď Aε,for all non-negative y. We also have |Fεpxq| ď 1 for all real x. Thus gives us a bound for F onthe positive x and y exes. Moreover, we certainly can find B so that

|Fεpzq| ď AεepT`εqp|z|´yq ď Aεe2pT`εq|z| ď Be|z|32 .

We can therefore apply the Phdagmén–Lindelöf theorem with β “ 32ă 2 “ α and obtain

|Fεpzq| ď maxtAε, 1u :“ Afor all z “ x ` iy such that x ě 0 and y ě 0. If we now repeat this argument for the secondquadrant, we can apply the Phdagmén–Lindelöf theorem again to Fε on the upper half-plane andβ “ 0 ă 1 “ α to obtain |F px` iyq ď 1 for y ě 0. Letting εÑ 0 we obtain |F px` iyq| ď eTyfor y ě 0. The lemma is then established by applying the result to Gpzq “ F p´zq. �

40 XU WANG

Proof of 1q implies 2q. The main idea is to use the following identityż

R|Upx` iyq|2 dx “ sup |Gpiyq|2, Gpzq :“

ż

RUpz ` tqfptq dt,

where the supremum is taken over all smooth functions f with bounded support in R such thatş

R |fptq|2 dt “ 1. By the Schwartz inequality, we know that

|Gpxq|2 ďż

R|Upx` tq|2dt ¨

ż

R|fptq|2 dt “

ż

R|Uptq|2dt :“ A2

Apply the above lemma to F pzq :“ Gpzq{A (notice that 1q implies that F is of exponential type2πR), we get

|Gpx` iyq| ď e2πR|y|A.Thus 2q follows. �

Remark: For other proofs and related results, see Seip’s notes [11] or page 158–160 in [3]).

2.6. Sampling theorem. The Paley–Wiener theorem tells us that one can hear the support(bandwidth) of a signal by estimating the exponential order of its Fourier–Laplace transform. Inthis section, we shall study a sampling property of Fourier–Laplace transform of an L2-functionwith bounded support. The main result is the following„ see page 167 in [12].

Theorem 2.16. Let f P L2r´1{2, 1{2s. Identify f with a function on R supported on r´1{2, 1{2s.Then for every x P R, we have

f̂pxq “8ÿ

n“´8f̂pnqsin πpx´ nq

πpx´ nq ,

andż

R|f̂pxq|2 dx “

ÿ

nPZ|f̂pnq|2.

Proof. Since te2πinxu is an orthonormal basis of L2r´1{2, 1{2s, we know that f is the L2 limitof the following sequence of functions

fN :“ÿ

|n|ďN

pf, e2πinxqe2πiny,

as N goes to infinity. Notice that

pf, e2πinxq “ż

|x|ă1{2fpxqe´2πinx dx “

ż

Rfpxqe´2πinx dx “ f̂pnq.

Identity each fN as a function on R supported on r´1{2, 1{2s, we know that the limit of fN is fin L2pRq. Since Fourier transform preserves the L2pRq norm (Plancherel identity) we know thatf̂ is equal to the L2 limit of

xfNpxq “ÿ

|n|ďN

f̂pnqż

|y|ă1{2e´2πixye2πiny dy.

FOURIER ANALYSIS 41

Now integration by parts givesż

|y|ă1{2e´2πixye2πiny dy “ sin πpx´ nq

πpx´ nq ,

where we define the value at n of the right hand as 1. Thus the first formula follows. The secondfollows from

ż

R|f̂pxq|2 dx “

ż

|x|ă1{2|fpxq|2 dx “

ÿ

nPZ|f̂pnq|2.

�

Remark 1: If we first look at the Fourier series expansion of f in L2r´λ{2, λ{2s, λ ą 1,then for an arbitray continuous function χ such that χ “ 1 on r´1{2, 1{2s and χ “ 0 outsider´λ{2, λ{2s, we have

||f ´ χfN || “ ||χf ´ χfN || Ñ 0, N Ñ 8,which gives

||f̂ ´ yχfN || Ñ 0, N Ñ 8,in particular, if we choose χ such that χpxq is linear when 1{2 ă |x| ă λ{2, we will get thefollowing oversampling formula

f̂pxq “ÿ

nPZ

1

λf´n

λ

¯

Kλ

´

x´ nλ

¯

, Kλpyq :“cos πy ´ cos πλyπ2pλ´ 1qy2 .

Thus, if one samples f̂ "more often", the series in the above formula converges faster sinceKλpyq “ Op1{|y|2q as |y| Ñ 8.

Remark 2: The sampling theroem is essentially the Fourier tranform of the Fourier seriesexpansion. In fact, since the Fourier transform preserves the L2 inner product on R, and

te2πinx||x|ă1{2uis an orthonormal system in L2pRq, we know that it is Fourier transform

sin πpx´ nqπpx´ nq

(

is also an orthonormal system in L2pRq. Thus the samping formula is also an orthogonal decom-position formula.

2.7. Heisenberg uncertainty principle. See page 158–161 and 168–169 in [12].

The mathematical thrust of the principle can be formulated in terms of a relation between afunction and its Fourier transform. The basic underlying law, formulated in its vaguest and mostgeneral form, states that a function and its Fourier transform cannot both be essentially localized.Somewhat more precisely, if the "preponderance" of the mass of a func-tion is concentrated inan interval of length L, then the preponderance of the mass of its Fourier transform cannot lie inan interval of length essentially smaller than L´1. The exact statement is as follows.

42 XU WANG

Theorem 2.17. Assume that f P S satisfies the normalization conditionş

R |fpxq|2 dx “ 1. Then

ˆż

Rx2|fpxq|2 dx

˙

¨ˆż

Ry2|f̂pyq|2 dy

˙

ě 116π2

,

and equality holds iff fpxq “ Ae´Bx2 where B ą 0 and A2 “a

2B{π.

Proof. Step 1: Beginning with our normalizing assumptionş

R |fpxq|2 dx “ 1, and recalling that

both f and f 1 are rapidly decreasing, an integration by parts gives

1 “ż

R|f |2 dx “ ´

ż

Rx p|f |2q1 dx “ ´

ż

Rx pf 1f̄ ` ff̄ 1q dx.

Therefore

1 ď 2ż

R|x| ¨ |f | ¨ |f 1| dx ď 2||xf || ¨ ||f 1||,

where we have used the Cauchy–Schwartz inequality. The identities

||f 1||2 “ ||f̂ 1||2 “ ||2πiyf̂ ||2 “ 4π2||yf̂ ||2,

which hold because of the properties of the Fourier transform and the Plancherel formula, conl-cudes the proof of the inequality in the theorem.

Step 2: If equality holds then we must also have equality where we applied the Cauchy–Schwarz inequality, and as a result we find that f 1pxq “ cxfpxq for some constant c. Thesolutions to this equation are fpxq “ Aecx2{2, where A is constant. Since we want f P S, wemust take c “ ´2B ă 0, now ||f || “ 1 gives |A2| “

a

2B{π. �

Remark: Replacing f by e´2πixy0fpx` x0q and changing variables one only getˆż

Rpx´ x0q2|fpxq|2 dx

˙

¨ˆż

Rpy ´ y0q2|f̂pyq|2 dy

˙

ě 116π2

for every x0, y0 P R. The precise assertion contained in the above inequality first came to lightin the study of quantum mechanics. It arose when one considered the extent to which one couldsimultaneously locate the position and momentum of a particle. Assuming we are dealing with(say) an electron that travels along the real line, then according to the laws of physics, mattersare governed by a "state function" f , which we can assume to be in S, and which is normalizedaccording to the requirement that ||f || “ 1. The position of the particle is then determined notas a definite point x; instead its probable location is given by the rules of quantum mechanics asfollows:

The probability that the particle is located in the interval pa, bq isşb

a|fpxq|2 dx.

According to this law we can calculate the probable location of the particle with the aid of f :in fact, there may be only a small probability that the particle is located in a given interval pa1, b1q,but nevertheless it is somewhere on the real line since

ş

R |fpxq|2 dx “ 1.

FOURIER ANALYSIS 43

In addition to the probability density |fpxq|2 dx, there is the expectation of where the particlemight be. This expectation is the best guess of the position of the particle, given its probabilitydistribution determined by |fpxq|2 dx, and is the quantity defined by

(2.10) Epxq :“ż

Rx|fpxq|2 dx.

Why is this our best guess? Consider the simpler (idealized) situation where we are given that theparticle can be found at only finitely many different points, x1, x2, ¨ ¨ ¨ , xN on the real axis, withpi the probability that the particle is at xi, and p1` p2` ¨ ¨ ¨ ` pN “ 1. Then, if we knew nothingelse, and were forced to make one choice as to the position of the particle, we would naturallytake Epxq “

ř

xipi, which is the appropriate weighted average of the possible positions. Thequantity (2.10) is clearly the general (integral) version of this.

We next come to the notion of variance, which in our terminology is the uncertainty attachedto our expectation. Having determined that the expected position of the particle is Epxq (givenby (2.10)), the resulting uncertainty is the quantity

(2.11)ż

Rpx´ Epxqq2|fpxq|2 dx.

Notice that if f is highly concentrated near Epxq, it means that there is a high probability that xis near Epxq, and so (2.11) is small, because most of the contribution to the integral takes placefor values of x near Epxq. Here we have a small uncertainty. On the other hand, if fpxq is ratherflat (that is, the probability distribution |fpxq|2 dx is not very concentrated), then the integral(2.11) is rather big, because large values of px ´ Epxqq2 will come into play, and as a result theuncertainty is relatively large.

It is also worthwhile to observe that the expectation Epxq is that choice for which the uncer-tainty

ş

Rpx ´ Epxqq2|fpxq|2 dx is the smallest. Indeed, if we try to minimize this quantity by

equating to 0 its derivative with respect to Epxq, we find that ´2ş

Rpx ´ Epxqq|fpxq|2 dx “ 0,

which gives (2.10). So far, we have discussed the "expectation" and "uncertainty" related to theposition of the particle. Of equal relevance are the corresponding notions regarding its momen-tum. The corresponding rule of quantum mechanics is:

The probability that the momentum y of the particle belongs to the interval pa, bq isşb

a|f̂pyq|2 dy,

where f̂ is the Fourier transform of f .

Combining these two laws with Theorem 2.17 gives 1{16π2 as the lower bound for the productof the uncertainty of the position and the uncertainty of the momentum of a particle. So the morecertain we are about the location of the particle, the less certain we can be about its momentum,and vice versa. However, we have simplified the statement of the two laws by rescaling to changethe units of measurement. Actually, there enters a fundamental but small physical number ~called Planck’s constant. When properly taken into account, the physical conclusion is

uncertainty principle: (uncertainty of position)ˆ (uncertainty of momentum) ě ~{16π2.

2.8. Central limit theorem. See page 114–116 in [3]

44 XU WANG

2.9. Fast Fourier transform. See page 224–226 in [12]

3. WAVELET ANALYSIS

Filter theory and signals, applications, TBA

4. APPENDIX 1: DEFINITION OF e, π AND EULER’S FORMULA

4.1. Definition of e. Recall that: Let A : Cn Ñ Cn be a linear map (here linear map meansApau ` bvq “ aApuq ` bApvq for all a, b in C and all u, v in Cn). We call u ‰ 0 in Cn aneigenvector of A if

(4.1) Au “ λu,where λ is a constant in C.

What is an eigenvector of the derivative ?

By (4.1), fix a complex number λ, we want to find a function u : RÑ C such thatu1 “ λu.

Power series method: Assume thatupxq “ a0 ` a1x` a2x2 ` ¨ ¨ ¨ ` anxn ` ¨ ¨ ¨ .

Formally, we have

u1pxq “ a1 ` 2a2x` ¨ ¨ ¨ ` nanxn´1 ` pn` 1qan`1xn ` ¨ ¨ ¨and

u1 “ λuô λan “ pn` 1qan`1, n “ 0, 1, ¨ ¨ ¨ .Thus

an`1 “λanpn` 1q “

λ2an´1pn` 1qn “ ¨ ¨ ¨ “

λn`1a0pn` 1qn ¨ ¨ ¨ 1 “

λn`1a0pn` 1q! ,

where we definen! “ 1 ¨ 2 ¨ ¨ ¨n.

Then we have

upxq “ u0 ¨ p1` λx` ¨ ¨ ¨ `pλxqn

n!` ¨ ¨ ¨ q.

PutEpxq :“ 1` x` ¨ ¨ ¨ ` x

n

n!` ¨ ¨ ¨ .

Since for every C ą 0,limnÑ8

Cn

n!“ 0,

we know that Epxq converges for all x in C.Theorem 4.1. Epλxq is a unique solution of the eigenvalue equation

u1 “ λu,with initial condition up0q “ 1.

FOURIER ANALYSIS 45

Definition 4.1. We shall define

e :“ Ep1q “ 1` 1` 12` ¨ ¨ ¨ ` 1

n!` ¨ ¨ ¨ .

4.2. Definition of the exponential function. Let us write

e2 “ e ¨ e, e3 “ e2 ¨ e,and define em inductively by

en`1 “ en ¨ e.Since e is positive, we can take the q-th root of em, we write it as e

mq . Thus for every x P Q, ex

is well defined. The following lemma tells us that Epxq is an extension of ex from Q to C.Lemma 4.1. For every x P Q, we have ex “ Epxq.

Proof. Since Ep1q “ e, it suffices to prove(4.2) Epλ1qEpλ2q “ Epλ1 ` λ2q,for every λ1, λ2 in C. Notice that

pEpλ1xqEpλ2xqq1 “ Epλ1xq1Epλ2xq ` Epλ2xq1Epλ1xq.Put

Gpxq “ Epλ1xqEpλ2xq.Apply Epλxq1 “ λEpλxq, we get

G1 “ pλ1 ` λ2qG.Notice that Gp0q “ 1. Thus Theorem 4.1 implies that

Gpxq “ Eppλ1 ` λ2qxq.Take x “ 1, we get Epλ1qEpλ2q “ Epλ1 ` λ2q. �

Exercise: Find a direct proof of Epλ1qEpλ2q “ Epλ1 ` λ2q without using Theorem 4.1.Definition 4.2. We shall use the same symbol ex to denote Epxq for all x in C and call ex theexponential function.

Remark 1: By Theorem 4.1, we know that ex is fully determined by

pexq1 “ ex, e0 “ 1.

Remark 2: Notice that e ą 0, we know that ex ą 0 for all x P Q, thus Epxq ą 0 for all x P R(since Epxq is smooth and Q is dense in R). Now we have E 1pxq “ Epxq ą 0 for real x, whichimplies that Epxq is strictly increasing on R. Moreover, we have

limxÑ´8

Epxq “ 0, limxÑ8

Epxq “ 8,

which implies that E maps R onto p0,8q. Thus every x ą 0 has a unique preimage, say lnx,such that Eplnxq “ x.Definition 4.3. We call lnx, x ą 0 the natural logarithmic function.

46 XU WANG

Remark: We have elnx “ x for every x P R, moreover pexq1 “ ex gives

plnxq1 “ 1x, x ą 0.

4.3. Definition of π and trigonometric functions. : Fix P0 “ p1, 0q in the unit circle

S1 :“ tpx, yq P R2 : x2 ` y2 “ 1u.

A counterclockwise rotation of P0 gives a arc P0P . The length, say θpP q, of the arc P0P is afunction of P . It is clear that the circumference diameter ratio is equal to θp´1, 0q.Definition 4.4 (Definition of π). We shall write the circumference diameter ratio as π.

Denote byF : θpP q ÞÑ P,

the inverse function of 0 ď θpP q ď 2π.Definition 4.5. We shall write F pθq “ pcos θ, sin θq and call cos θ, sin θ the trigonometric func-tions.

Notice thatF p0q “ p1, 0q “ F p2πq, F pπq “ p´1, 0q, |F pθq| ” 1.

In particular, it gives

sinp0q “ sinp2πq “ 0, cosp0q “ cosp2πq “ 1.

By definition of θ, we have (since θ is the arclength parameter of S1)ż θ̂

0

|F 1pθq| dθ “ θ̂, 0 ď θ̂ ď 2π,

which gives|F 1pθq| ” 1.

Now F pθq ¨ F pθq ” 1 implies

F 1 ¨ F ` F ¨ F 1 “ 2F ¨ F 1 ” 0.

Hence F 1KF , thus we know that

F 1pθq “ p´ sin θ, cos θq, or F 1pθq “ psin θ,´ cos θq.

But notice that F 1p0q “ p0, 1q, thus we must have

F 1pθq “ p´ sin θ, cos θq,

which is equivalent to (here we use i2 “ ´1)

pcos θ ` i sin θq1 “ ipcos θ ` i sin θq.

Notice that cos 0` i sin 0 “ 1, thus Theorem 4.1 givesTheorem 4.2 (Euler’s formula). eiθ “ cos θ ` i sin θ.

FOURIER ANALYSIS 47

Take θ “ π, we get the following Euler’s identityeiπ “ ´1.

Moreover, apply (4.2), we geteiθ1eiθ2 “ eipθ1`θ2q,

thus by Euler’s formula, we have

pcos θ1 ` i sin θ1qpcos θ2 ` i sin θ2q “ cospθ1 ` θ2q ` i sinpθ1 ` θ2q,i.e.

(4.3) cospθ1 ` θ2q “ cos θ1 cos θ2 ´ sin θ1 sin θ2,and

(4.4) sinpθ1 ` θ2q “ sin θ1 cos θ2 ` cos θ1 sin θ2.

5. APPENDIX 2: LEBESGUE INTEGRAL

We shall follow section 1.1 in [3], the readers are recommened to read [13] for the motivations.

5.1. Lebesgue integral on r´π, πs.Definition 5.1 (Lebesgue measure). The class of Lebesgue measurable sets in r´π, πs is thesmallest collection, say L, of subsets of r´π, πs such that

1) ra, bs P L for all ´π ď a ă b ď π;2) E P L if |E| :“ inf

ř8n“1 lengthpInq “ 0, where the infimum is taken over the class of

countable coverings of E by means of open intervals In (i e Y8n“1In Ą E);3) L is closed under countable unions, countable intersections and complementation.

We call |E| the Lebesgue measure of E if E is Lebesgue measurable.Remark: Without the second assumption, the above definition gives Borel measurable sets.

Definition 5.2 (measurable function). We call a real function f : r´π, πs Ñ R a measurablefunction if f´1ppa, bsq are Lebesgue measurable for all real numbers a ă b.

Remark: A complex function is said to be Lebesgue measurable if both its real and imaginaryparts are Lebesgue measurable. We shall denote by Mr´π, πs the space of all complex Lebesguemeasurable functions.Definition 5.3 (Lebesgue integral of a nonnegative measurable function). Let f : r´π, πs Ñr0,8q be a measurable function. We call

ż π

´πfpxq dx :“ lim

nÑ8

8ÿ

k“0k2´n

ˇ

ˇf´1pAkqˇ

ˇ, Ak :“ pk2´n, pk ` 1q2´ns,

the Lebesgue integral of f . We say that f is integrable ifşπ

´π fpxq dx ă 8.Remark: A general real measurable function f is integrable if both maxtf, 0u and maxt´f, 0u

are integrable. A complex measurable function is integrable if both its real and imaginary partsare integrable.

48 XU WANG

Now let us define the Lp space p ě 1. First we say that f “ g almost everywhere in r´π, πs if|tf ‰ gu| “ 0.Definition 5.4. Fix p ě 1, we shall define

Lpr´π, πs :“ tf PMr´π, πs :ż π

´π|f |p dx ă 8u{ „,

wheref „ g ô f “ g a.e. on r´π, πs.

5.2. Lebesgue measure on Rn. Just replace interval by n times product of intervals. We onlygive the definition of Lebesgue measurable sets in Rn and leave the other definitions to thereaders.Definition 5.5. The class of Lebesgue measurable sets in Rn is the smallest collection, say L, ofsubsets of Rn such that

1) L contains all closed n-cubes;2) E P L if |E| :“ inf

ř8n“1 lengthpInq “ 0, where the infimum is taken over the class of

countable coverings of E by means of open n-cubes In (i e Y8n“1In Ą E);3) L is closed under countable unions, countable intersections and complementation.

We call |E| the Lebesgue measure of E if E is Lebesgue measurable.

Example: Lebesgue integral of e´φ:

Ipφq :“ż

Rne´φpxq dx1 ¨ ¨ ¨ dxn,

where φpxq is a real Lebesgue measurable function.Theorem 5.1. Assume that for every real number s, |Ωpsq| ă 8, where

Ωpsq :“ tx P Rn : φpxq ă su.Then we have

Ipφq “ż 8

´8|Ωpsq| e´s ds.

Proof. By definition of the Lebesgue integral, we have

Ipφq “ limnÑ8

8ÿ

k“0k2´n|tk2´n ă e´φ ď pk ` 1q2´nu|.

Since each |Ωpsq| is finite, we can write |tk2n ă e´φ ď pk ` 1q2nu| as|Ωp´ logpk2´nqq| ´ |Ωp´ logppk ` 1q2´nqq|.

Thus the above limit of sums can be written as (try!)

Ipφq “ limnÑ8

8ÿ

k“12´n|Ωp´ logpk2´nqq|.

FOURIER ANALYSIS 49

Notice thatt ÞÑ |Ωp´ log tq|, t ą 0,

is a positive locally bounded decreasing function on 0 ă t ă 8, thus it is Riemann integrable(see Proposition 1.3 in [12]). We know that the above limit is equal to the following Riemannintegral

ż 8

0

|Ωp´ log tq| dt.

Thus our formula follows by a change of variable t “ e´s. �Remark: We can also get a similar formula for a general Lebesgue measurable set in Rn (not

only on Rn). For example, if a function is well defined and negative on a Lebesgue measurableset Ω, extend φ such that φ “ 8 outside Ω then

Ωpsq “ Ω, @ s ě 0.Thus the above theorem gives

Ipφq “ż 0

´8|Ωpsq| e´s ds` |Ω| ¨

ż 0

´8e´s ds.

Notice thatş0

´8 e´s ds “ 1, hence we have

ż

Ω

e´φpxq dx1 ¨ ¨ ¨ dxn “ż 0

´8|Ωpsq| e´s ds` |Ω|,

in case φ ă 0 in Ω.

6. EXERCISE SETS

6.1. Exercise set 1: Fejér kernel and its applications. — From page 53–58, page 63 in [12],see also page 34–36 in [3].

6.1.1. Fejér’s theorem. The aim is to prove the following theorem:

Theorem 6.1 (Fejér’s theorem). If f P PC0pS1q then

limNÑ8

ˇ

ˇ

f0pxq ` ¨ ¨ ¨ ` fN´1pxqN

´ fpx`q ` fpx´q2

ˇ

ˇ “ 0,

By definition we have (see (1.2))

fNpx0q “ pf,DNpx´ x0qq,where the Dirichlet kernel DNpxq is defined by

DNpxq :“Nÿ

n“´Nωn, ω :“ eix.

Thus we havef0px0q ` ¨ ¨ ¨ ` fN´1px0q

N“ pf, FNpx´ x0qq,

50 XU WANG

where

FNpxq :“řN´1n“0 Dnpxq

N,

is called the Fejér kernel. We have proved that (see the proof of (1.1))

Dnpxq “ω´n ´ ωn`1

1´ ω ,

Exercise 1: Prove the following facts on the Fejér kernel

1) FNpxq “ 1Nsin2pNx{2qsin2px{2q ;

2) 12π

şπ

´π FNpxq dx “ 1.

Put

INpxq :“f0pxq ` ¨ ¨ ¨ ` fN´1pxq

N´ fpx`q ` fpx´q

2.

Use Exercise 1 to prove that:

Exercise 2: Assume further that f is continuous at x0. Then

1) INpx0q “ 12πşπ

´π pfpxq ´ fpx0qq1N

sin2pNx{2qsin2px{2q dx;

2) For every 0 ă δ ă π, we have limNÑ8ş

δď|x|ďπ1N

sin2pNx{2qsin2px{2q dx “ 0;

3) limNÑ8 INpx0q “ 0.

Exercise 3: Use Exercise 2 and the method in section 1.4.4 to prove