Construction of wavelets through Walsh functions

Preface......Page 6
 References......Page 9
Contents......Page 11
About the Authors......Page 14
Introduction......Page 16
 References......Page 19
 1.1 Walsh Functions......Page 22
 1.2 Walsh–Fourier Transform......Page 26
 1.3 Haar Functions and Its Relationship with Walsh Functions......Page 29
 1.4 Walsh-Type Wavelet Packets......Page 31
  1.5.1 Continuous Wavelet Transform......Page 33
  1.5.2 Discrete Wavelet System......Page 36
  1.5.3 Multiresolution Analysis......Page 37
 1.6 Wavelets with Compact Support......Page 38
 1.7 Exercises......Page 44
 References......Page 45
 2.1 Walsh–Fourier Coefficients......Page 47
  2.1.1 Estimation of Walsh–Fourier Coefficients......Page 48
  2.1.2 Transformation of Walsh–Fourier Coefficients......Page 51
 2.2 Convergence of Walsh–Fourier Series......Page 54
  2.2.1 Summability in Homogeneous Banach Spaces......Page 59
  2.3.1 Approximation by Césaro Means of Walsh–Fourier Series......Page 60
  2.3.2 Approximation by Nörlund Means of Walsh–Fourier Series in Lp Spaces......Page 65
  2.3.3 Approximation by Nörlund Means in Dyadic Homogeneous Banach Spaces and Hardy Spaces......Page 77
  2.4.1 Image Representation and Transmission......Page 90
  2.4.2 Data Compression......Page 93
  2.4.5 ECG Analysis......Page 94
  2.4.6 EEG Analysis......Page 95
  2.4.8 Pattern Recognition......Page 96
 References......Page 97
 3.1 Haar System and Its Generalization......Page 99
 3.2 Haar Fourier Series......Page 102
 3.3 Haar System as Basis in Function Spaces......Page 106
 3.4 Non-uniform Haar Wavelets......Page 108
 3.5 Generalized Haar Wavelets and Frames......Page 113
  3.6.1 Applications to Solutions of Initial and Boundary Value Problems......Page 114
  3.6.2 Applications to Solutions of Integral Equations......Page 116
 3.7 Exercises......Page 117
 References......Page 118
 4.1 Preliminary......Page 119
 4.2 Orthogonal Wavelets and MRA in L2(mathbbR+)......Page 123
 4.3 Orthogonal Wavelets with Compact Support on mathbbR+......Page 134
 4.4 Estimates of the Smoothness of the Scaling Functions......Page 153
 4.5 Approximation Properties of Dyadic Wavelets......Page 178
 4.6 Exercise......Page 189
 References......Page 190
 5.1 Multiresolution Analysis on Vilenkin Groups......Page 191
 5.2 Compactly Supported Orthogonal p-Wavelets......Page 214
 5.3 Periodic Wavelets on Vilenkin Groups......Page 225
 5.4 Periodic Wavelets Related to the Vilenkin–Christenson Transform......Page 231
 5.5 Application to the Coding of Fractal Functions......Page 246
 References......Page 254
 6.1 Introduction......Page 255
  6.2.1 Haar–Vilenkin Mother Wavelet......Page 256
 6.3 Approximation by Haar–Vilenkin Wavelets......Page 263
 6.4 Covergence Theorems......Page 266
 6.5 Haar–Vilenkin Coefficients......Page 268
 6.6 Exercises......Page 271
 References......Page 272
 7.1 Biorthogonal Wavelets on R+......Page 273
 7.2 Biorthogonal Wavelets on Vilenkin Groups......Page 283
 7.3 Construction of Biorthogonal Wavelets on The Vilenkin Group......Page 289
 7.4 Frames on Vilenkin Group......Page 292
 7.5 Application to Image Processing......Page 297
 References......Page 301
 8.1 Introduction......Page 303
 8.2 Nonuniform Multiresolution Analysis on Positive Half Line......Page 305
  8.2.1 Construction of Nonuniform Multiresolution Analysis on Positive Half Line......Page 311
  8.2.2 The Analogue of Cohen\'s Condition......Page 317
 8.3 Nonuniform Wavelet Frames in L2(mathbbR)......Page 323
  8.3.1 Necessary and Sufficient Condition......Page 327
 8.4 Exercises......Page 334
 References......Page 335
 9.1 Introduction......Page 336
 9.2 Vector-Valued Multiresolution Analysis......Page 337
 9.3 Vector-Valued Multiresolution p-Analysis on mathbbR+......Page 340
 9.4 The Existence of Orthogonal Vector-Valued Wavelets on mathbbR+......Page 345
 9.5 Vector-Valued Nonuniform Multiresolution Analysis......Page 353
  9.5.1 Construction of Vector-Valued Nonuniform Multiresolution Analysis......Page 362
 9.6 Vector-Valued Nonuniform Wavelet Packets  and Its Properties......Page 366
  9.6.1 Vector-Valued Nonuniform Wavelet Bases......Page 374
 9.7 Exercise......Page 376
 References......Page 377
 A.1 Basic Results of Functional Analysis......Page 378
 A.2 Vilenkin Systems......Page 386
  A.3.1  The Dyadic Group G......Page 388
 A.4 Introduction to Real and Fourier Analysis......Page 391
 References......Page 397
Lake Index......Page 398
Index......Page 399