상세정보

Advances in Shannon's Sampling Theory provides an up-to-date discussion of sampling theory, emphasizing the interaction between sampling theory and other branches of mathematical analysis, including the theory of boundary-value problems, frames, wavelets, multiresolution analysis, special functions, and functional analysis. The author not only traces the history and development of the theory, but also presents original research and results that have never before appeared in book form. Recent techniques covered include the Feichtinger-Grochenig sampling theory; frames, wavelets, multiresolution analysis and sampling; boundary-value problems and sampling theorems; and special functions and sampling theorems. The book will interest graduate students and professionals in electrical engineering, communications, and applied mathematics.

Advances in Shannon's Sampling Theory provides an up-to-date discussion of sampling theory, emphasizing the interaction between sampling theory and other branches of mathematical analysis, including the theory of boundary-value problems, frames, wavelets, multiresolution analysis, special functions, and functional analysis.

정보제공 :

CONTENTS
1. INTRODUCTION AND A HISTORICAL OVERVIEW = 1
 1.1 A Historical Overview = 1
 1.2 Introduction and Terminology = 4
 References = 12
2 SHANNON SAMPLING THEOREM AND BAND-LIMITED SIGNALS = 15
 2.0 Introduction = 15
 2.1 Shannon Sampling Theorem and the Cardinal Series = 15
  2.1.A Shannon Sampling Theorem = 15
  2.1.B The Cardinal Series and Whittaker's Cardinal Function = 18
  2.1.C Oversampling = 22
 2.2 More Band-Limited Functions = 23
 References = 35
3. GENERALIZATIONS OF SHANNON SAMPLING THEOREMS = 37
 3.0 Introduction = 37
 3.1 Non-Uniform Sampling = 38
 3.2 Sampling Theorems for Other Integral Transforms and Representations of Band-Limited Signals - Kramer's Sampling Theorem = 45
 3.3 Multidimensional Sampling = 56
 3.4 Sampling Theorems and Generalized Functions = 62
 3.5 Sampling Theorems for Other Types of Signals = 64
  3.5.A Time-Limited Signals (Nonband-Limited Signals) = 64
  3.5.B Band-Pass Signals = 66
  3.5.C Finite Power Signals = 68
 3.6 Sampling by Using More General Types of Data and Sampling Function = 74
  3.6.A Sampling by Using Other Types of Data = 74
  3.6.B Sampling by Using Other Types of Sampling Functions = 81
 3.7 Sampling Theorems in More General Function Spaces = 83
 3.8 Error Analysis = 84
  3.8.A Truncation Errors = 85
  3.8.B Aliasing Errors = 92
  3.8.C Amplitude Errors = 93
  3.8.D Time-Jitter Errors = 94
  3.8.E Combined Errors = 95
 3.9 Closing Remark = 96
 References = 98
4. SAMPLING THEOREMS ASSOCIATED WITH STURMLIOUVILLE BOUND ARY-VALUE PROBLEMS = 107
 4.0 Introduction = 107
 4.1 The Regular Case = 108
 4.2 The Singular Case on a Halfline = 115
 4.3 The Singular Case on the Whole Line = 122
 4.4 Examples = 130
  4.4.A The Regular Case = 130
  4.4.B The Singular Case on a Halfline = 131
  4.4.C The singular Case on the Whole Line = 134
  4.4.D One-Sided Cardinal Series Associated with Linear Forms of Eigenvalues = 138
 4.5 Counter-example = 145
 4.6 Open Question = 147
 References = 149
5. SAMPLING THEOREMS ASSOCIATED WITH SELFADJCINT BOUNDARY-VALUE PROBLEMS = 153
 5.0 Introduction = 153
 5.1 Preliminaries = 154
 5.2 The Sampling Theorem = 163
 References = 166
6. SAMPLING BY USING GREEN'S FUNCTION = 167
 6.0 Introduction = 167
 6.1 Preliminaries = 168
 6.2 The Sampling Theorem = 172
 6.3 The Haddad-Yao-Thomas Sampling Theorem = 182
 References = 185
7. SAMPLING THEOREMS AND SPECIAL FUNCTIONS = 187
 7.0 Introduction = 187
 7.1 Historical Background = 187
 7.2 New Summation Formulae Involving Trigonometric and Bessel Functions = 191
 7.3 Summation Formulae Involving the Zeros of the Bessel Function = 196
 7.4 Summation Formulae Involving the Γ-Function = 199
 7.5 Summation Formula Involving the Kamp$$\acute e$$ de F$$\acute e$$riet Function = 201
 References = 203
8. KRAMER'S SAMPLING THEOREM AND LAGRANGETYPE INTERPOLATION IN N DIMENSIONS = 205
 8.0 Introduction = 205
 8.1 Preliminaries = 205
 8.2 An N-Dimensional Kramer's Sampling Theorem = 206
 8.3 Examples = 212
 8.4 Applications to Special Functions = 217
 8.5 Open Questions = 228
 References = 229
9. SAMPLING THEOREMS FOR MULTIDIMENSIONAL SIGNALS-THE FEICHTINGER-GR$$\ddot O$$CHENIG SAMPLING THEORY = 231
 9.0 Introduction = 231
 9.1 The Feichtinger - Gr$$\ddot o$$chenig Sampling Theory = 232
 9.2 Sampling by Using the Green's Function in Several Variables = 243
 References = 248
10. FRAMES AND WAVELETS : A NEW PERSPECTIVE ON SAMPLING THEOREMS = 249
 10.0 Introduction = 249
 10.1 Sampling in Reporducing-Kernel Hilbert Spaces = 252
 10.2 Frames and Sampling Theorems = 259
  10.2.A Frames = 259
  10.2.B The Discrete Windowed Fourier Transform Functions and The Gabor Frames = 266
  10.2.C The Zak Transform = 274
  10.2.D Sampling Theorems by Using Frames = 280
 10.3 Wavelet Analysis and Sampling = 287
  10.3.A Wavelets = 287
  10.3.B The Wavelet Frame Operator = 298
  10.3.C Multiresolution Analysis = 300
  10.3.D Wavelets and Sampling = 317
 References = 324
Author Index = 327
Subject Index = 329