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Fourier Analysis

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发表于 2014-9-4 15:18:07 | 显示全部楼层 |阅读模式
电子图书
电子图书名: Fourier Analysis
编者: E Stein, R Shakarchi
内容简介: 该书是美国普林斯顿大学的E Stein教授所著,E Stein教授是菲尔兹奖获得者陶哲轩的老师,本书由浅入深的讲述了傅立叶变换原理。从事信号或谐波分析的学者或研究人员可以参考。。。
所属专业方向: 谐波分析、信号检测等
出版社: Princeton University Press
来源: 网络

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preface 9 h2 G7 K- q2 M
chapter 1. the genesis of fourier analysis . k; j/ ?0 r) x4 _) x
1 the vibrating string
( \% i$ ]: n2 G4 m% _% \0 R1.1 derivation of the wave equation
) o: [+ ?& K3 F2 W4 s1.2 solution to the wave equation
0 R, b+ B5 F7 @* N2 E3 e1.3 example: the plucked string
0 Z9 K& n  ]9 g" K9 ~. w, ~: a2 the heat equation
% t% {2 U& ~. T9 w# U2.1 derivation of the heat equation : Q9 \6 ]' h. k; |' t8 e9 g
2.2 steady-state heat equation in the disc 6 _" ?0 r% j8 n
3 exercises
1 Q: w/ Y. {( d/ S$ f4 problem 5 }* ]- Z5 s( H6 o+ @  K! F3 q2 h

4 f3 T! g( c4 D6 c! k, ]chapter 2. basic properties of fourier series
1 n* I4 T9 f, f/ m% E. z6 K% j; e; N1 examples and formulation of the problem 2 f4 d( D6 m' e  ?
1.1 main definitions and some examples # g) \# x) u8 a! e
2 uniqueness of fourier series 4 t$ f3 N1 a5 m+ j
3 convolutions 0 w5 F  q* ?9 R. q& y8 n8 e" ?( e
4 good kernels
$ p5 w1 @, C/ e% i% `' U5 cesaro and abel summability: applications to fourierseries
9 E1 m6 ]% Z: @$ _0 F$ V: n9 P.5.1 cesaro means and snmmation , X! {4 R, e' a0 [5 F0 l8 {
5.2 fejer's theorem 2 A8 F; c6 A4 k
5.3 abel means and s-ruination
  x- \& X( ~- n5.4 the poisson kernel and dirichlet's problem in the unitdisc
9 {$ D* S, v% x( X4 s2 A6 exercises
& n( B$ t, }8 @/ H+ C4 j2 H+ A& ^7 problems # a4 k3 n8 I$ k
3 {5 {9 v  T! {+ L# p( A5 t2 L) t
chapter 3. convergence of fourier series & b& @" p) }5 G! D9 h7 [. Q3 ~; O
1 mean-square convergence of fourier series
; f; E( P* A  E! E* D2 z1.1 vector spaces and inner products
& _: q+ |! Y3 a6 R' G- a$ k1.2 proof of mean-square convergence
* s: ~$ F- c0 ]- v; A4 A2 return to pointwise convergence
) P! o; i* q4 t* y/ b2.1 a local result 8 q% S- u) \  v' M8 D; p1 M
2.2 a continuous function with diverging fourierseries
& \+ g/ Q9 X. `2 l/ }# t3 exercises
; T' w: [9 n9 W6 ?1 e2 v+ ~& ^4 problems ( f" d% O: U- [' s! d
5 B5 s8 j, r) J" p& C, ~/ d
chapter 4. some applications of fourier series 3 N9 b  g" T! s
1 the isoperimetric inequality
" M% E# L8 J: p+ h2 i0 k  V2 weyl's equidistribution theorem $ q' {" g' o. z/ o/ `1 q
3 a continuous but nowhere differentiable function
2 Q! F% r  i4 s7 |4 the heat equation on the circle
4 m8 a( @) _. M2 w9 {+ u* a1 Z& h5 exercises
1 w6 n2 f3 z4 u) ?, y, U( k6 problems
- r6 \6 u7 S' Q $ P0 e( f. Q/ z& a
chapter 5. the fourier transform on r / |" s$ _: @4 Y: }3 k( O: L/ J
1 elementary theory of the fourier transform 4 |; j) Z. |3 x, s/ x2 H6 V9 a1 h
1.1 integration of functions on the real line . ~$ Z( C" i. S8 t
1.2 definition of the fourier transform 6 I: H& F8 r' O
1.3 the schwartz space 3 `/ T8 w0 I( m) P' z) ^3 q
1.4 the fourier transform on 3
8 i: c' f, m$ F  V1 B; o2 Z3 A! E1.5 the fourier inversion
! j. t1 i5 ?) d6 j: f1.6 the plancherel formula
9 D% e! M+ a( c" u9 h- }: R1.7 extension to functions of moderate decrease   M7 x/ y9 y' o( f$ _
1.8 the weierstrass approximation theorem
! s& {% c# v1 n3 E2 applications to some partial differential equations : f2 ^# g2 Y% o& ^" t
2.1 the time-dependent heat equation on the real line - p! p* P& }1 v1 o' Q' ^
2.2 the steady-state heat equation in the upperhalf-plane / d0 _# z( q3 |' X8 M* V0 c- M
3 the poisson summation formula
2 b6 r" _- W! @& p7 `3.1 theta and zeta functions
8 J! y* Q: ~4 @! ?; t. w3.2 heat kernels + _4 ]# t+ Y; S* f0 i
3.3 poisson kernels
/ _1 S, C9 t& w: W' o& L4 the heisenberg uncertainty principle
* `9 y; {& h  V% q4 A' I2 f5 exercises ) v% p8 n2 n: `
6 problems
9 D- c$ l7 `, |9 X: v; T ) s* @( D( V1 m& b1 X8 b
chapter 6. the fourier transform on ra " Q9 @% l  w* b, U7 r' _: [5 I5 n
1 preliminaries 9 {9 y0 l/ D: J3 ]
1.1 symmetries
' \$ I& X, g, G1.2 integration on ra
; c$ x; [8 l5 h5 t4 }, w7 e2 elementary theory of the fourier transform ( g1 q2 r5 L: C6 N' [
3 the wave equation in rd ×r : c; f* {" @) c
3.1 solution in terms of fourier transforms
2 q: h# z9 m( n8 m0 [3.2 the wave equation in r3× r
5 D# d: z9 k% {/ B* M3.3 the wave equation in r2 × r: descent 7 @) r( t5 j; Q
4 radial symmetry and bessel functions , j6 ~; g7 J$ r9 v) ?
5 the radon transform and some of its applications
; `& }( L2 f8 ~) X% F3 u( E5.1 the x-ray transform in r2
( q5 k$ o& r  W* l5.2 the radon transform in r3
4 {5 O; T% o1 b5.3 a note about plane waves ; l2 E" w2 @5 G# u5 L7 s
6 exercises
$ S; |* u" ]" S. \- s) V5 b7 problems : `$ d& z6 w( X7 I5 f* J, Q/ E( f
6 R3 P3 ]  d# T) c7 s( H
chapter 7. finite fourier analysis
$ t" M: e$ J7 R3 l5 g1 fourier analysis on z(n)
5 b$ w% Z6 c- j' B1.1 the group z(n)
1 L- v8 _& _$ {) q2 j" c# b3 Y1.2 fourier inversion theorem and plancherel identity onz(n) 5 W2 n7 {2 H+ x  L: H: G, ^
1.3 the fast fourier transform 7 B5 C1 x, z, Z' X' m
2 fourier analysis on finite abelian groups
% j' m+ q. @$ \2 R2.1 abelian groups
8 z1 z, U8 y6 b0 ?/ g8 c; e* K4 t2.2 characters
, C0 r) `& i4 e4 P2 o" f- a2.3 the orthogonality relations
9 ^; S$ k  ^! v" g6 e7 r( v9 H2.4 characters as a total family
5 |! V1 |: R  Z1 \% P2 {- m$ i2.5 fourier inversion and plancherel formula
  l: M6 `; V# a* U/ h& Q3 exercises * p4 s) M- E% q4 d( T. o- {
4 problems 7 S( A% o9 \+ I) d  G* G2 q' U
! [* Y" o% g& n) a# j
chapter 8. dirichlet's theorem
. u2 m6 e- m3 c! V# `, D1 a little elementary number theory 2 o1 x' a. S: b& K$ i$ U
1.1 the fundamental theorem of arithmetic
) a. t8 v) j8 k1.2 the infinitude of primes
' d; H% `8 ?4 C2 dirichlet's theorem
6 w* d$ @9 M! A: L2.1 fourier analysis, dirichlet characters, and reduc-tion ofthe theorem
' c8 m; {) M6 a# Z5 b) {$ d* w2.2 dirichlet l-functions
1 t7 o' y2 i! W, B% h# R3 proof of the theorem
6 b& d: V$ ]$ ~# B* n" m* r3.1 logarithms # r' g$ ]; e& o3 C
3.2 l-functions ; x$ a6 ]$ w  ^* c- t
3.3 non-vanishing of the l-function ! f/ O1 D* A* U- |6 d# Z1 L
4 exercises
3 \8 K2 Y, `) n" z5 problems ) F8 n. U8 L8 F6 b7 N, M1 u, z
appendix: integration
' D2 C2 v5 T& b3 i8 D9 q1 definition of the riemann integral
% K( k4 o7 e( y/ X- c1.1 basic properties ; z8 C- l" z" p" }& r$ E
1.2 sets of measure zero and discontinuities of inte-grablefunctions
4 M( h! a5 B" z/ i/ }1 H( \1 c) X2 multiple integrals
/ e4 p& k$ O- x" A" v/ a! V$ b3 I2.1 the riemann integral in rd
% F. E* o: I* A2 q; W2.2 repeated integrals
1 q! g# h; r2 f9 \2.3 the change of variables formula
! y4 u9 J. C% G1 E) v2.4 spherical coordinates 5 R( ?# ?6 p7 l/ p6 q. i! O. ]5 b( S
3 improper integrals. integration over rd 1 ]( |1 j1 e; t+ |/ Z% o0 i
3.1 integration of functions of moderate decrease
( J: ?! G7 y( x3.2 repeated integrals
. I$ f5 b- J; O! q3.3 spherical coordinates
2 K/ |) _2 E+ d% rnotes and references
+ I* E) E1 d/ b- K2 m1 o# q' [3 b$ kbibliography * E0 X. u7 ^, l, @: }# s+ n
symbol glossary
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