This book is intended as an introduction to classical Fourier analysis, Fourier series, and the Fourier transform. The topics are developed slowly for the reader who has never seen them before, with a preference for clarity of exposition in stating and proving results. More recent developments, such as the discrete and fast Fourier transforms and wavelets, are covered in the last two chapters. The first three, short, chapters present requisite background material, and these could be read as a short course in functional analysis. The text includes many historical notes to place the material in a cultural and mathematical context; from the fact that Jean Baptiste Joseph Fourier was the nineteenth, but not the last, child in his family to the impact that Fourier series have had on the evolution of the concept of the integral.
Author(s): George Bachmann, Lawrence Narici, Edward Beckenstein
Series: Universitext
Publisher: Springer
Year: 2000
Language: English
Pages: 515
City: New York
Cover......Page 1
Title page......Page 3
Date-line......Page 4
Foreword......Page 5
Contents......Page 7
1.1 Metric Spaces......Page 11
1.2 Normed Spaces......Page 19
1.3 Inner Product Spaces......Page 22
1.4 Orthogonality......Page 28
1.5 Linear Isometry......Page 34
1.6 Holder and Minkowski Inequalities; $L_p$ and $l_p$ Spaces......Page 38
2.1 Balls......Page 45
2.2 Convergence and Continuity......Page 48
2.3 Bounded Sets......Page 59
2.4 Closure and Closed Sets......Page 62
2.5 Open Sets......Page 68
2.6 Completeness......Page 70
2.7 Uniform Continuity......Page 76
2.8 Compactness......Page 79
2.9 Equivalent Norms......Page 85
2.10 Direct Sums......Page 93
3 Bases......Page 99
3.1 Best Approximation......Page 100
3.2 Orthogonal Complements and the Projection Theorem......Page 104
3.3 Orthonormal Sequences......Page 113
3.4 Orthonormal Bases......Page 117
3.5 The Haar Basis......Page 124
3.6 Unconditional Convergence......Page 129
3.7 Orthogonal Direct Sums......Page 133
3.8 Continuous Linear Maps......Page 136
3.9 Dual Spaces......Page 141
3.10 Adjoints......Page 145
4 Fourier Series......Page 149
4.1 Warmup......Page 153
4.2 Fourier Sine Series and Cosine Series......Page 164
4.3 Smoothness......Page 169
4.4 The Riemann-Lebesgue Lemma......Page 179
4.5 The Dirichlet and Fourier Kernels......Page 184
4.6 Pointwise Convergence of Fourier Series......Page 198
4.7 Uniform Convergence......Page 212
4.8 The Gibbs Phenomenon......Page 217
4.9 A Divergent Fourier Series......Page 220
4.10 Termwise Integration......Page 223
4.11 Trigonometric vs. Fourier Series......Page 228
4.12 Termwise Differentiation......Page 231
4.13 Dido's Dilemma......Page 234
4.14 Other Kinds of Summability......Page 236
4.15 Fejer Theory......Page 245
4.16 The Smoothing Effect of $(C,1)$ Summation......Page 252
4.17 Weierstrass's Approximation Theorem......Page 254
4.18 Lebesgue's Pointwise Convergence Theorem......Page 255
4.19 Higher Dimensions......Page 259
4.20 Convergence of Multiple Series......Page 267
5 The Fourier Transform......Page 273
5.1 The Finite Fourier Transform......Page 274
5.2 Convolution on $\mathbf{T}$......Page 277
5.3 The Exponential Form of Lebesgue's Theorem......Page 283
5.4 Motivation and Definition......Page 285
5.5 Basics/Examples......Page 288
5.6 The Fourier Transform and Residues......Page 294
5.7 The Fourier Map......Page 299
5.8 Convolution on $\mathbf{R}$......Page 301
5.9 Inversion, Exponential Form......Page 304
5.10 Inversion, Trigonometric Form......Page 305
5.11 $(C,1)$ Summability for Integrals......Page 313
5.12 The Fejer-Lebesgue Inversion Theorem......Page 316
5.13 Convergence Assistance......Page 327
5.14 Approximate Identity......Page 340
5.15 Transforms of Derivatives and Integrals......Page 344
5.16 Fourier Sine and Cosine Transforms......Page 350
5.17 Parseval's Identities......Page 361
5.18 The $L_2$ Theory......Page 366
5.19 The Plancherel Theorem......Page 371
5.20 Pointwise Inversion and Summability......Page 376
5.21 A Sampling Theorem......Page 382
5.22 The Mellin Transform......Page 385
5.23 Variations......Page 388
6.1 The Discrete Fourier Transform......Page 393
6.2 The Inversion Theorem for the DFT......Page 400
6.3 Cyclic Convolution......Page 406
6.4 Fast Fourier Transform for $N=2^k$......Page 409
6.5 The Fast Fourier Transform for $N=RC$......Page 416
7 Wavelets......Page 421
7.1 Orthonormal Basis from One Function......Page 423
7.2 Multiresolution Analysis......Page 424
7.3 Mother Wavelets Yield Wavelet Bases......Page 429
7.4 From MRA to Mother Wavelet......Page 432
7.5 Construction of a Scaling Function with Compact Support......Page 445
7.6 Shannon Wavelets......Page 458
7.7 Riesz Bases and MRAs......Page 459
7.8 Franklin Wavelets......Page 469
7.9 Frames......Page 474
7.10 Splines......Page 486
7.11 The Continuous Wavelet Transform......Page 490
Index......Page 507
