preface e( I h9 j0 g) a# Kchapter 1. the genesis of fourier analysis 2 w/ I% x" r: ?: P1 the vibrating string , @1 N+ M- h6 `: d* U/ L
1.1 derivation of the wave equation - B& I2 v0 m: ~: N7 q
1.2 solution to the wave equation 6 u V+ @, O8 [9 r! \: A. p) C
1.3 example： the plucked string 1 B8 Q9 S4 f5 i* r8 O$ [2 the heat equation r' S1 F4 V5 S0 S. Q/ T8 P2.1 derivation of the heat equation ' v( P$ d, t! o8 }8 A7 W3 f e1 A2.2 steady-state heat equation in the disc 0 B4 F8 W4 A. L A0 ^8 @3 exercises : q8 d+ _+ p) @$ H* | y8 r4 problem * T' C! L& b: O. c' z: @( a* l% ^ # {$ M' a% X$ }chapter 2. basic properties of fourier series ( Q6 b$ l: ~% v8 @& ~& U" }1 examples and formulation of the problem / ?8 |0 N; j4 H( g( I1.1 main definitions and some examples & _( k9 i- d3 l, _* O2 uniqueness of fourier series & n( a( U7 i" z3 convolutions / c( C) Q- T/ \" x3 N6 Q( G# {) Q+ Q
4 good kernels % K$ I# `: f1 C2 s- f. _5 cesaro and abel summability： applications to fourierseries 6 Y! v. R5 _& s
.5.1 cesaro means and snmmation 3 P, y1 i: s& c8 J- r# T7 U5.2 fejer's theorem . ~2 n6 ]2 C9 \1 u& r5.3 abel means and s-ruination ( H; D) G4 H! d0 A/ b, l5.4 the poisson kernel and dirichlet's problem in the unitdisc ' j8 C2 a" E i; d5 T' _7 x
6 exercises 7 d2 |; j5 ^9 y3 P
7 problems 0 e5 g/ K- H( H7 b ' r' i1 b: I T1 y4 q9 m" U
chapter 3. convergence of fourier series , B% M; R R( M& P8 W1 mean-square convergence of fourier series 6 e6 j7 B/ X) s" E& h5 m. l
1.1 vector spaces and inner products . ?5 r! N/ ~0 V4 W
1.2 proof of mean-square convergence 1 U5 P0 v. Z# }5 Z1 j$ \0 w2 return to pointwise convergence 0 u6 a5 \% L: i1 R5 p+ F' b: G& I2.1 a local result * m1 m _" h2 I3 d- F8 c+ M
2.2 a continuous function with diverging fourierseries * k. Z7 a5 {8 ?" W8 f; E: q3 exercises . j) e% `: W: B, }7 ?9 Q0 ~( m, d4 problems . ~! Y# @4 V' L# D7 l 0 B" r( D Q( y5 {
chapter 4. some applications of fourier series + [" B2 x4 I+ ~* H) ]
1 the isoperimetric inequality 4 ?) W$ v: Q3 `* x, ^6 a2 weyl's equidistribution theorem 9 O2 E) V! ?3 u3 z
3 a continuous but nowhere differentiable function 8 v8 Z! i0 J% @" K4 l7 p/ R
4 the heat equation on the circle ( c5 l5 b. P6 c: h/ A" O/ \) d
5 exercises 7 H4 x4 E0 F! d6 problems " N0 }7 B- Q' n9 ]) ` / Z+ z. |2 ?5 G! b7 \) Z! B
chapter 5. the fourier transform on r # d# g7 K! X& ?' b3 [( a
1 elementary theory of the fourier transform 4 u- m% y W. G6 g% R" j& T' Z& n/ }0 ^
1.1 integration of functions on the real line u5 X; O8 F: ]1.2 definition of the fourier transform 3 ^7 w n" E; L8 R
1.3 the schwartz space # S" s M8 W$ s* y1.4 the fourier transform on 3 * z! y2 P' o/ z# a7 a8 v
1.5 the fourier inversion 5 m |, B, N. Y
1.6 the plancherel formula + P; J4 Z% e( v
1.7 extension to functions of moderate decrease 3 x) `9 ]7 [. Y$ H# S. h) y8 y
1.8 the weierstrass approximation theorem 6 o4 \5 i% m3 T' J/ A7 i2 applications to some partial differential equations : T* _" S' N4 s
2.1 the time-dependent heat equation on the real line / t' s# g! V0 z1 d
2.2 the steady-state heat equation in the upperhalf-plane # \3 f3 Y/ O. H5 d
3 the poisson summation formula . a4 T2 v* e! L- h
3.1 theta and zeta functions , N0 p3 G4 f, \% u& ~$ _/ p3.2 heat kernels 0 F* F6 U$ Q1 i2 @2 D# |% e+ h( Q3.3 poisson kernels ' u' Q, p' ~! f- V# J( u
4 the heisenberg uncertainty principle % u' B! [" Q3 ]$ v) R# w+ c
5 exercises 0 k6 {- a/ ^7 f9 P
6 problems 8 C1 [0 I! r7 C, U, j
( p: `6 T1 A8 W5 E- Y% X* }. J8 achapter 6. the fourier transform on ra 3 y# Y9 ~) i& C; ?( d1 preliminaries 9 P) L6 [3 F4 ]1 y/ ^
1.1 symmetries * b% I# O% T+ b& Q1.2 integration on ra # ~' [0 W M6 s; W* z1 ?, ?9 C7 Y
2 elementary theory of the fourier transform & y# U5 P" b$ ~3 {# G- r3 the wave equation in rd ×r 5 A" o/ K; K3 c, a6 \% `3.1 solution in terms of fourier transforms $ P& e% G& \5 Z4 h. W/ W9 R
3.2 the wave equation in r3× r # C* `. | C8 j2 `1 T3.3 the wave equation in r2 × r： descent - Y0 P" i g- u: q A
4 radial symmetry and bessel functions & z) s5 U' p( i; `) Z8 ~+ S
5 the radon transform and some of its applications , Z( k# J3 {; y8 m5.1 the x-ray transform in r2 5 l. S8 a* N' X2 p. n. y5.2 the radon transform in r3 1 N" d' s' E4 [! e. a5.3 a note about plane waves 6 B; G7 Q: |8 b- Z. A8 e
6 exercises / i" _: M8 D6 `% I4 S: d$ N5 ?7 l
7 problems 6 H& _" G: c6 {- M3 K+ s. \' }# s
) o/ _1 {5 o5 a4 x' @7 c1 G
chapter 7. finite fourier analysis 5 X6 Q/ j; D- O2 @/ c1 fourier analysis on z（n） : K* W5 i# \! V! ?* k, f1.1 the group z（n） ) ]& `0 H& P: [# I8 }8 g: g1.2 fourier inversion theorem and plancherel identity onz（n） % u4 j- @% u8 h2 w8 [' K0 W
1.3 the fast fourier transform 8 M. P( {. L- _6 Y$ w! ^. ^( _
2 fourier analysis on finite abelian groups 9 N3 i. k( @7 N# `+ i+ w/ {
2.1 abelian groups 0 {% N% m0 n0 w8 P2.2 characters . |! {8 D8 I3 F1 M7 m2.3 the orthogonality relations 5 o! j0 l9 W V" P/ j8 M4 v2.4 characters as a total family , d. D' z# A9 o! k# f5 u3 s" z2.5 fourier inversion and plancherel formula 4 q: z/ v+ b. _ Y0 P; n
3 exercises ! h0 j6 L- s% l( E4 problems # E& L' P0 X0 E/ T8 Z/ J" t/ B
- `" s' J0 t4 ]0 {chapter 8. dirichlet's theorem * P9 {9 f: @, ^, O7 i; A- |
1 a little elementary number theory 4 _! t: A. R$ `) S5 l1.1 the fundamental theorem of arithmetic " o. w: u% p$ M+ r5 \9 Y8 [' H
1.2 the infinitude of primes . [6 v. `; I) p y* \
2 dirichlet's theorem , w7 O; A# }5 u
2.1 fourier analysis， dirichlet characters， and reduc-tion ofthe theorem % X1 V, G9 M9 m d+ ~
2.2 dirichlet l-functions , E) D( O D* z
3 proof of the theorem 8 R+ j$ f3 ^9 V! L# H; m$ ^% r( i3.1 logarithms , H: x% k9 x A3.2 l-functions # ?+ C# c( O* p/ i" Q3.3 non-vanishing of the l-function . ~: J, S5 e) ~7 \
4 exercises ' i% \2 t1 y7 Y z; h( B5 problems % {, x7 R4 t! e# Rappendix： integration $ D) X, F! X0 o$ O G2 b* _
1 definition of the riemann integral * f. }$ m8 V! {0 ^% n
1.1 basic properties 1 l& z) Q* q8 B1 u+ R- C1.2 sets of measure zero and discontinuities of inte-grablefunctions ; \6 s7 @" d- R, c
2 multiple integrals & l7 G7 z( k% R% F& Z' S$ ?) e$ a2.1 the riemann integral in rd 2 J- F( N+ B. K
2.2 repeated integrals h: [# U7 u, A1 v5 b" z
2.3 the change of variables formula q% O) m, C y. Y" |2.4 spherical coordinates 2 V. n" n8 |7 n# e& _/ ^8 V# |# I3 improper integrals. integration over rd / k* \. ^$ U- k* K4 y; l7 `3.1 integration of functions of moderate decrease 4 y1 Z! \5 [0 o7 a# u9 d |
3.2 repeated integrals 9 Z& T! S5 Q3 ?) n
3.3 spherical coordinates 8 a2 j- e. S& n j6 {+ Ynotes and references ( W" k; v. L. Q' z& v, h0 ?7 Fbibliography * Z, r; [1 Y1 \& J' }* {, osymbol glossary
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