Landscapes of Time-Frequency Analysis - ATFA 2019

ANHA Series Preface......Page 7
Preface......Page 10
References......Page 12
Acknowledgements......Page 13
Contents......Page 14
Contributors......Page 18
1 Introduction......Page 20
2.1 Notation......Page 23
2.2 Setting and Assumptions......Page 24
2.3 The Unitarization Theorem and Inversion Formula......Page 27
3 Dual Pairs and Irreducibility......Page 28
3.1 Irreducibility......Page 30
4.1.1 Groups and Spaces......Page 36
4.1.2 The Representations......Page 38
4.1.3 The Radon Transform......Page 39
4.1.4 The Unitarization Theorem......Page 41
4.2.1 Groups and Spaces......Page 42
4.2.3 The Radon Transform......Page 43
4.2.4 The Unitarization Theorem......Page 45
References......Page 46
1 Introduction and Main Result......Page 48
2 An Isometric Isomorphism......Page 51
Appendix......Page 57
Appendix......Page 60
References......Page 61
1 Introduction......Page 62
1.1 Notation......Page 64
2 Preliminaries......Page 65
2.1 The Spaces......Page 66
2.3 The Affine Radon Transform......Page 69
3 The Shearlet Transform......Page 70
4 The Shearlet Synthesis Operator......Page 74
Appendix......Page 77
Appendix......Page 79
References......Page 81
1 Introduction......Page 82
2.1 Modulation Spaces......Page 84
2.2 Main Results on L2......Page 87
3 Eigenvalues and Eigenfunctions......Page 88
4 Gelfand–Shilov Spaces Framework......Page 89
4.2 Ultra-Modulation Spaces......Page 90
4.3 Boundedness and Schatten Class......Page 91
References......Page 92
1 Introduction......Page 94
1.2 Why Do We Need a Time-Varying Spectrum Theory?......Page 95
2 The Joint Density Approach......Page 96
3.1 Uncertainty Principle Argument......Page 97
3.1.2 Classical Examples of Joint Distributions with an Uncertainty Principle......Page 98
3.5 Uncertainty Product Argument......Page 100
3.6 Time and Frequency Resolution ``Trade Off''......Page 101
3.7 The Uncertainty Principle for the Spectrogram......Page 102
4.1 Bilinear Distributions Satisfying the Marginals......Page 103
4.2 Manifestly Positive Bilinear Densities Not Satisfyingthe Marginals......Page 104
4.3 Non-bilinear, Manifestly Positive Distributions Satisfying the Marginals......Page 105
5 Why Is There an Exception to Wigner's Theorem?......Page 107
6 Is Time-Frequency ``Hidden'' in Standard Fourier Analysis?......Page 109
7 Can One Derive Time-Frequency Densities?......Page 110
7.1 A New Approach......Page 112
8 What Did Wigner Do in His 1932 Paper?......Page 113
8.1 The Wigner Distribution as a Tool for Solving DifferentialEquations......Page 114
References......Page 117
1 Introduction......Page 121
2 Preliminary Definitions and Facts......Page 123
3 Distribution Multipliers......Page 129
4 Unbounded Distribution Multipliers......Page 132
5 Riesz Distribution Multipliers......Page 136
6 Conclusions......Page 138
References......Page 139
1.1 Motivations......Page 141
1.2 Notation and Terminology......Page 143
2 Density Operators: Summary......Page 144
3 Toeplitz Operators: Definitions and Properties......Page 146
4 Toeplitz Quantum States......Page 148
5 Discussion and Perspectives......Page 151
References......Page 152
1 Introduction: The Planck Constant .12em.1emdotteddotteddotted.76dotted.6h Is for What?......Page 153
2.1 Fourier Analysis, Like in Euclidean Geometry…......Page 156
2.2 Gabor Signal Analysis ( Time-Frequency)......Page 157
2.3 Continuous Wavelet Transform ( Time-Scale)......Page 158
3.2 Probabilistic Content of Integral Quantization: Semi-Classical Portraits......Page 159
3.3 Classical Limit......Page 160
4.1 PV Measures for Quantization: Time Operator......Page 161
4.3 Mutatis Mutandis…and CCR......Page 162
5.1 From PV Quantization to Gabor POVM Quantization......Page 163
5.2 Beyond Gabor Quantization......Page 164
6 Affine Quantization......Page 167
7 Examples of Operatorial Signal Analysis Through Gabor Quantization......Page 169
8 Discussion......Page 172
References......Page 173
1 Introduction......Page 174
2.1 Anatomic Structure of the Retina......Page 176
2.2 Retinal Manifestations of Disease......Page 177
2.4 Image Dataset......Page 178
3 Automated Image Analysis of Retinal Vascularization......Page 179
3.1 Local Measures of Retinal Vascularization......Page 180
3.2 Global Measures of Retinal Vascularization......Page 181
3.3 Segmentation of Retinal Vessels......Page 184
References......Page 186
1 Introduction......Page 192
2 A Few Facts on Modulation Spaces......Page 195
3.1 The Sequential Approach......Page 197
3.2 The Time-Slicing Approximation......Page 199
4 Beyond the L2 Theory via Gabor Analysis......Page 202
4.1 The Role of Modulation Spaces......Page 203
4.2 Sparse Operators on Phase Space......Page 204
5 Higher Order Rough Parametrices......Page 206
5.1 The Role of .12em.1emdotteddotteddotted.76dotted.6h......Page 209
6 Pointwise Convergence of Integral Kernels......Page 210
6.1 Weyl Operators......Page 211
6.2 Main Results......Page 212
6.3 The Proof at a Glance......Page 215
6.4 Why Exceptional Times?......Page 216
References......Page 217
Applied and Numerical Harmonic Analysis (101 volumes)......Page 220