Fourier Analysis

控制理论

preface
chapter 1. the genesis of fourier analysis
% |9 D5 z8 c$ o$ I6 s4 E1 the vibrating string
1.1 derivation of the wave equation
  Z% W3 ~; x- R; y2 J- C( X2 r9 G) _1.2 solution to the wave equation
5 H7 \! z7 k  r4 i1.3 example： the plucked string
2 the heat equation
2.1 derivation of the heat equation
& M' |: m$ `2 N2.2 steady-state heat equation in the disc
& k: G0 N7 k! `( W8 e2 w' h) J3 exercises
4 problem

chapter 2. basic properties of fourier series
  x( C5 f) x1 B5 A1 examples and formulation of the problem
1.1 main definitions and some examples
2 uniqueness of fourier series
) I0 T8 u; d4 m# P' K3 convolutions
4 good kernels
: g: w. \3 C" D5 q5 cesaro and abel summability： applications to fourierseries
$ U/ y" h3 H) n/ v/ f1 ^.5.1 cesaro means and snmmation
5.2 fejer's theorem
5.3 abel means and s-ruination
& f; O6 b. O0 ^5.4 the poisson kernel and dirichlet's problem in the unitdisc
6 exercises
7 problems
" s- g; X: ]  h" d0 @% t
; Z" P) j. |2 d* ~chapter 3. convergence of fourier series
2 E6 M! p+ I% y( [1 mean-square convergence of fourier series
1.1 vector spaces and inner products
( h+ B/ ]6 W& [. s; O- A1.2 proof of mean-square convergence
2 return to pointwise convergence
1 G8 k* _8 S) w1 l; n7 A) z6 H' [2.1 a local result
2.2 a continuous function with diverging fourierseries
' B- @4 }' p3 C) N* I% r3 exercises
& d/ b1 x/ K- ^  E! \; T4 problems

1 E8 V' D3 B. K% K7 y: l8 Zchapter 4. some applications of fourier series
: o  `9 y" T* v# u) F2 Z1 the isoperimetric inequality
/ p# Y- w) ]9 |/ M2 weyl's equidistribution theorem
3 a continuous but nowhere differentiable function
$ B" D* B3 m" E: r# C- {) F4 the heat equation on the circle
) R" Q; B% h5 n' x5 exercises
9 ^8 B; z) Z. |# K6 problems
6 Q$ c' U% M% B% c3 w
chapter 5. the fourier transform on r
1 elementary theory of the fourier transform
5 f, l% {( j) D0 T1.1 integration of functions on the real line
1.2 definition of the fourier transform
. u  L8 `  a! {1 m1.3 the schwartz space
1.4 the fourier transform on 3
1.5 the fourier inversion
1.6 the plancherel formula
' Y3 `+ H" B2 M7 t& {9 r+ l1.7 extension to functions of moderate decrease
  ]  d* l* c7 W  U+ E1.8 the weierstrass approximation theorem
# V3 M: ?9 U" t6 m2 applications to some partial differential equations
2.1 the time-dependent heat equation on the real line
2.2 the steady-state heat equation in the upperhalf-plane
8 j( Q# ^" Z8 C0 J9 l3 the poisson summation formula
: z/ i4 f/ e9 S$ |2 ^& U3.1 theta and zeta functions
3.2 heat kernels
' c- B- l$ [; x3.3 poisson kernels
+ |* H3 m% ?0 b8 S5 ]2 o% K, |4 the heisenberg uncertainty principle
5 exercises
6 problems
' L* r1 O2 y/ t6 u7 @
7 Z- ^. N5 e. J4 q8 c; L$ r1 M  Achapter 6. the fourier transform on ra
, d5 `2 f# J4 U% H1 preliminaries
3 x- |. n+ q  q. Z5 n$ O% G0 Q1.1 symmetries
1.2 integration on ra
2 elementary theory of the fourier transform
3 the wave equation in rd ×r
3.1 solution in terms of fourier transforms
3.2 the wave equation in r3× r
- k* o) j3 {. i$ Z( v" A- k3.3 the wave equation in r2 × r： descent
& h1 z2 Q4 c- d+ ?7 F4 radial symmetry and bessel functions
: k% r/ Z3 c5 K0 r$ `2 A1 d/ c5 the radon transform and some of its applications
" ~1 l% C3 |0 r5.1 the x-ray transform in r2
5.2 the radon transform in r3
6 Y* K8 w4 ]+ _3 ^6 O5.3 a note about plane waves
6 exercises
! S0 S0 ]/ i( m( L6 g7 problems

chapter 7. finite fourier analysis
9 {0 N7 w1 j2 c$ o1 ~& ]1 [, n1 fourier analysis on z（n）
1.1 the group z（n）
1.2 fourier inversion theorem and plancherel identity onz（n）
+ X* l1 K7 j7 z  S1.3 the fast fourier transform
2 fourier analysis on finite abelian groups
6 i% G2 S1 e- a- O7 u& U! C2.1 abelian groups
2.2 characters
2.3 the orthogonality relations
2.4 characters as a total family
2.5 fourier inversion and plancherel formula
3 exercises
4 problems

chapter 8. dirichlet's theorem
1 a little elementary number theory
1.1 the fundamental theorem of arithmetic
! h8 b+ ^7 R1 m8 G3 E1.2 the infinitude of primes
0 ~6 m& @; W& Z9 M: T" d  U' U2 dirichlet's theorem
2.1 fourier analysis， dirichlet characters， and reduc-tion ofthe theorem
2.2 dirichlet l-functions
% e: h$ j; C. j% u5 s, Q3 proof of the theorem
3.1 logarithms
$ w+ r8 Z0 E2 q( }4 Z8 P4 j3.2 l-functions
3.3 non-vanishing of the l-function
4 exercises
2 J' B( G8 Z9 Z% ?+ i, x+ n! |$ Y5 problems
appendix： integration
1 definition of the riemann integral
1.1 basic properties
1.2 sets of measure zero and discontinuities of inte-grablefunctions
2 multiple integrals
2.1 the riemann integral in rd
2.2 repeated integrals
( X- n) s- P. |3 g. n# ]# [2.3 the change of variables formula
2.4 spherical coordinates
3 improper integrals. integration over rd
) Y& B0 I2 y5 Z+ X2 m% y; q: o/ R: w( b3.1 integration of functions of moderate decrease
% a: S7 S) s: g3 d( s7 b4 @) t$ E3.2 repeated integrals
" j/ o0 C6 ~5 ?/ }: y# S3.3 spherical coordinates
notes and references
bibliography
symbol glossary

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