Fourier Analysis

控制理论

preface
chapter 1. the genesis of fourier analysis
1 the vibrating string
* C) C; S  B8 n2 I1.1 derivation of the wave equation
1.2 solution to the wave equation
1.3 example： the plucked string
2 the heat equation
& p: n9 y! P+ a2.1 derivation of the heat equation
2 T0 K9 B6 \! c+ M2.2 steady-state heat equation in the disc
4 S# ~" R. I+ ~- Q  D. }8 B3 exercises
4 problem

" M; J, u3 y; z$ [8 {4 @6 Rchapter 2. basic properties of fourier series
7 N- y7 f( c0 W8 [3 a, [) S- O. C; F1 examples and formulation of the problem
) R5 Q) p3 s9 t# n2 b1.1 main definitions and some examples
' l9 ]& d- v4 o8 k0 T6 Q7 p2 uniqueness of fourier series
# ^8 M* W" i1 V. j3 convolutions
4 good kernels
5 cesaro and abel summability： applications to fourierseries
" e  G/ W7 y4 i6 A6 i% |.5.1 cesaro means and snmmation
2 K4 K, l9 `" _& W. A% s- V5.2 fejer's theorem
; e5 P* O* p& P3 [5.3 abel means and s-ruination
3 J9 ]' u, J2 u4 N# l3 @# `  W: W5.4 the poisson kernel and dirichlet's problem in the unitdisc
6 exercises
1 w3 x# @7 _. `2 b1 K- m( I( B7 problems

chapter 3. convergence of fourier series
1 mean-square convergence of fourier series
  ~9 l5 ^; g, B* O* X1.1 vector spaces and inner products
# x+ f8 t  H1 q' e, ]& D1 Z) r9 M1.2 proof of mean-square convergence
* r: C% D, d, B2 return to pointwise convergence
2.1 a local result
: z( C- ^1 M+ ^2 {2.2 a continuous function with diverging fourierseries
3 exercises
8 x. h+ k) k4 W! g4 problems
: [& T) g: Y9 r
; H" z6 _2 z& ?" m3 qchapter 4. some applications of fourier series
+ y8 U- m/ j+ B  N( s# b1 the isoperimetric inequality
# T( o- l1 g8 u9 H" w4 F6 }/ y2 weyl's equidistribution theorem
" R- ]1 c4 F: ^3 a continuous but nowhere differentiable function
; n, d. U6 g& x$ F7 e" e4 the heat equation on the circle
$ j/ o9 @. U8 c# D- V5 exercises
6 problems
( I, R0 ]9 a6 y3 `, z
! s0 X9 d, ~: j# |- [/ a9 Zchapter 5. the fourier transform on r
1 elementary theory of the fourier transform
1.1 integration of functions on the real line
1 f; f. C9 E4 i7 z% G! b. W1.2 definition of the fourier transform
7 s# z  y7 I0 }5 e1.3 the schwartz space
1 z$ m# t4 A; F. Q/ I1.4 the fourier transform on 3
1.5 the fourier inversion
1.6 the plancherel formula
1.7 extension to functions of moderate decrease
8 F- |* x% y! g5 }3 f8 J1.8 the weierstrass approximation theorem
, r0 V" b- l* R2 applications to some partial differential equations
: N( A' u9 L) U# A, S, }2.1 the time-dependent heat equation on the real line
2.2 the steady-state heat equation in the upperhalf-plane
3 the poisson summation formula
3.1 theta and zeta functions
# f) }$ b. m) d1 K$ L3.2 heat kernels
3.3 poisson kernels
' {1 s5 [0 i$ Z' y) Y2 K4 the heisenberg uncertainty principle
. B' O4 c" x6 f6 F% C$ i9 p; `% }5 exercises
6 problems

, \6 ^% Y5 Y) w1 r9 A' g# s0 }chapter 6. the fourier transform on ra
- [" m  r9 R5 P2 @- O+ {6 k, c/ F1 preliminaries
6 j- y, _# |+ ^( {! }8 i! l6 J; }1.1 symmetries
1.2 integration on ra
2 elementary theory of the fourier transform
4 l: w: N, i+ \" I! {* l: a3 the wave equation in rd ×r
3.1 solution in terms of fourier transforms
- b+ e' v- w& \9 o3.2 the wave equation in r3× r
8 S. h/ j. Y) N5 Q6 Q3.3 the wave equation in r2 × r： descent
4 radial symmetry and bessel functions
- Q* q/ ~9 e2 ]% y# V6 N5 the radon transform and some of its applications
5.1 the x-ray transform in r2
5.2 the radon transform in r3
2 \( _) h: i  W! k9 _- ~5.3 a note about plane waves
6 exercises
7 problems
1 J: M1 }2 d  w: g7 j
) K8 ]3 ?7 v* g5 t7 k( |chapter 7. finite fourier analysis
1 fourier analysis on z（n）
1.1 the group z（n）
1.2 fourier inversion theorem and plancherel identity onz（n）
4 |/ B% n* t& J& I4 G9 F1.3 the fast fourier transform
2 fourier analysis on finite abelian groups
. m9 R: b* K0 o/ b2.1 abelian groups
2.2 characters
2.3 the orthogonality relations
2.4 characters as a total family
2.5 fourier inversion and plancherel formula
( o2 ]5 N: ~( {( j  P1 N3 exercises
* x- C. c( G! _# f1 l! |4 problems

chapter 8. dirichlet's theorem
; N8 U4 ?* r1 `( q0 R/ N1 a little elementary number theory
1.1 the fundamental theorem of arithmetic
1.2 the infinitude of primes
2 dirichlet's theorem
0 e* s" @  F* Z* W' H! l2.1 fourier analysis， dirichlet characters， and reduc-tion ofthe theorem
2.2 dirichlet l-functions
. A2 x' s/ p! x# `" j+ {/ U* k3 proof of the theorem
7 V0 A6 {, w: h; v9 L3.1 logarithms
3.2 l-functions
; B! o; p' `( R6 ~2 w8 g" P0 b9 ]3.3 non-vanishing of the l-function
/ e  ?- c" D7 `6 Z( F' `4 exercises
5 problems
& ?4 ]) B% h7 T! yappendix： integration
* f$ a& \; I4 c/ w1 definition of the riemann integral
: `! b4 }4 ~9 O, R9 x  `1.1 basic properties
7 L5 l& e4 L/ Z/ }1.2 sets of measure zero and discontinuities of inte-grablefunctions
2 multiple integrals
2 [; g' S0 B" Q9 w( r2.1 the riemann integral in rd
* P7 i/ z: y0 v! C2.2 repeated integrals
2.3 the change of variables formula
2.4 spherical coordinates
3 improper integrals. integration over rd
3 r& O- n1 |& W3.1 integration of functions of moderate decrease
: L) s6 ]) ]- v& v0 [3.2 repeated integrals
3.3 spherical coordinates
notes and references
, Z' Z% v5 E  \: `bibliography
! M$ @) r! W: Y: r- p6 e/ fsymbol glossary

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