Lattice Point Identities and Shannon-Type Sampling

325 Pages·2019·3.396 MB·\325

by  Willi Freeden (Author), M. Zuhair Nashed (Author)

Lattice Point Identities
and Shannon-Type
Sampling

Lattice Point Identities
and Shannon-Type
Sampling
Willi Freeden
M. Zuhair Nashed
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Contents
Preface xi
Authors xv
Acknowledgments xvii
I Central Theme 1
1 From Lattice Point to Shannon-Type Sampling Identities 3
1.1 Classical Framework of Shannon Sampling . . . . . . . . . . 3
1.2 Transition From Shannon to Shannon-Type Sampling . . . . 7
1.3 Novel Framework of Shannon-Type Sampling . . . . . . . . . 8
2 Obligations, Ingredients, Achievements, and Innovations 11
2.1 Obligations and Ingredients . . . . . . . . . . . . . . . . . . . 11
2.2 Achievements and Innovative Results . . . . . . . . . . . . . 13
2.3 Methods and Tools . . . . . . . . . . . . . . . . . . . . . . . 14
3 Layout 17
3.1 Structural Organisation . . . . . . . . . . . . . . . . . . . . . 17
3.2 Relationship to Other Monographs . . . . . . . . . . . . . . . 21
II Univariate Poisson-Type Summation Formulas
and Shannon-Type Sampling 23
4 Euler/Poisson-Type Summation Formulas and
Shannon-Type Sampling 25
4.1 Classical Euler Summation Formula . . . . . . . . . . . . . . 25
4.2 Variants of the Euler Summation Formula . . . . . . . . . . 30
4.3 Poisson-Type Summation Formula over Finite Intervals . . . 33
4.4 Shannon Sampling Based on the Poisson Summation-Type
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
v
vi Contents
4.5 Shannon-Type Sampling Based on Poisson Summation-Type
Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.6 Fourier Transformed Values–Based Shannon-Type Sampling
(Finite Intervals) . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.7 Functional Values–Based Shannon-Type Sampling (Finite
Intervals) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.8 Paley–Wiener Reproducing Kernel Hilbert Spaces . . . . . . 47
4.9 Poisson-Type Summation Formula over the Euclidean Space 48
4.10 Functional Values–Based Shannon-Type Sampling (Euclidean
Space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.11 Fourier Transformed Values–Based Shannon-Type Sampling
(Euclidean Space) . . . . . . . . . . . . . . . . . . . . . . . . 56
III Preparatory Material for Multivariate Lattice
Point Summation and Shannon-Type Sampling 59
5 Preparatory Tools of Vector Analysis 61
5.1 Cartesian Notation and Settings . . . . . . . . . . . . . . . . 61
5.2 Spherical Notation and Settings . . . . . . . . . . . . . . . . 63
5.3 Regular Regions and Integral Theorems . . . . . . . . . . . . 65
6 Preparatory Tools of the Theory of Special Functions 71
6.1 Homogeneous Harmonic Polynomials . . . . . . . . . . . . . 71
6.2 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . 90
7 Preparatory Tools of Lattice Point Theory 93
7.1 Lattices in Euclidean Spaces . . . . . . . . . . . . . . . . . . 93
7.2 Figure Lattices in Euclidean Spaces . . . . . . . . . . . . . . 96
7.3 Basic Results of the Geometry of Numbers . . . . . . . . . . 98
7.4 Lattice Points Inside Spheres . . . . . . . . . . . . . . . . . . 100
8 Preparatory Tools of Fourier Analysis 109
8.1 Stationary Point Asymptotics . . . . . . . . . . . . . . . . . 110
8.2 Periodic Polynomials and Fourier Expansions . . . . . . . . . 113
8.3 Fourier Transform over Euclidean Spaces . . . . . . . . . . . 115
8.4 Periodization and Classical Poisson Summation Formula . . 117
8.5 Gauss–Weierstrass Transform over Euclidean Spaces . . . . . 119
8.6 Hankel Transform and Discontinuous Integrals . . . . . . . . 122
Contents vii
IV Multivariate Euler-Type Summation Formulas
over Regular Regions 127
9 Euler–Green Function and Euler-Type Summation
Formula 129
9.1 Euler–Green Function . . . . . . . . . . . . . . . . . . . . . . 129
9.2 Euler-Type Summation Formulas over Regular Regions
Based on Euler–Green Functions . . . . . . . . . . . . . . . . 131
9.3 Iterated Euler–Green Function . . . . . . . . . . . . . . . . . 134
9.4 Euler-Type Summation Formulas over Regular Regions
Based on Iterated Euler–Green Functions . . . . . . . . . . . 136
V Bivariate Lattice Point/Ball Summation and
Shannon-Type Sampling 141
10 Hardy–Landau-Type Lattice Point Identities (Constant
Weight) 143
10.1 Integral Mean Asymptotics for the Euler–Green Function . . 143
10.2 Hardy–Landau-Type Identity . . . . . . . . . . . . . . . . . . 146
10.3 Discrepancy Asymptotics . . . . . . . . . . . . . . . . . . . . 151
11 Hardy–Landau-Type Lattice Point Identities
(General Weights) 153
11.1 Pointwise Fourier Inversion Formula for Regular Regions . . 156
11.2 General Geometry and Homogeneous Boundary Weight . . . 158
11.3 Circles and General Weights . . . . . . . . . . . . . . . . . . 163
11.4 Smooth Convex Regions and General Weights . . . . . . . . 168
12 Bandlimited Shannon-Type Sampling (Preparatory
Results) 171
12.1 From Hardy–Landau-Type Identities to Shannon-Type
Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
12.2 Over- and Undersampling . . . . . . . . . . . . . . . . . . . . 172
13 Lattice Ball Euler Summation Formulas and
Shannon-Type Sampling 175
13.1 Lattice Ball Euler–Green Function . . . . . . . . . . . . . . . 175
13.2 Lattice Ball Euler Summation Formula . . . . . . . . . . . . 177
13.3 Lattice Ball Mean Shannon-Type Sampling . . . . . . . . . . 179
viii Contents
VI Multivariate Poisson-Type Summation
Formulas over Regular Regions 185
14 Gauss–Weierstrass Mean Euler-Type Summation
Formulas and Shannon-Type Sampling 187
14.1 Gauss–Weierstrass Transform over Regular Regions . . . . . 187
14.2 Gauss–Weierstrass Mean Euler–Green Function . . . . . . . 189
14.3 Gauss–Weierstrass Mean Euler Summation over Regular
Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
14.4 Bandlimited Gauss–Weierstrass Shannon-Type Sampling . . 191
15 From Gauss–Weierstrass to Ordinary Lattice Point
Poisson–Type Summation 193
15.1 Theta Function and Functional Equation . . . . . . . . . . . 193
15.2 Poisson-Type Summation over Regular Regions
(Gauss–Weierstrass Approach) . . . . . . . . . . . . . . . . . 196
15.3 Poisson-Type Summation over Regular Regions
(Ordinary Approach) . . . . . . . . . . . . . . . . . . . . . . 199
VII Multivariate Shannon-Type Sampling
Formulas over Regular Regions 203
16 Shannon-Type Sampling Based on Poisson-Type
Summation Formulas 205
16.1 Fourier-Transformed Values–Based Shannon-Type Sampling
(Gauss–Weierstrass Approach) . . . . . . . . . . . . . . . . . 206
16.2 Parseval-Type Identity (Gaussian/Ordinary Approach) . . . 212
16.3 Fourier-Transformed Values–Based Shannon-Type Sampling
(Ordinary Approach) . . . . . . . . . . . . . . . . . . . . . . 217
16.4 Functional Values–Based Shannon-Type Sampling (Gaussian
Approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
17 Paley–Wiener Space Framework and Spline
Approximation 223
17.1 Paley–Wiener Reproducing Kernel Structure . . . . . . . . . 223
17.2 Spline Interpolation in Paley–Wiener Spaces . . . . . . . . . 226
17.3 Paley–Wiener Spline Interpolatory Sampling . . . . . . . . . 230
17.4 Paley–Wiener Spline Interpolatory Cubature . . . . . . . . . 231
17.5 Multivariate Antenna Problem . . . . . . . . . . . . . . . . . 232
Contents ix
VIII Multivariate Poisson-Type Summation
Formulas over Euclidean Spaces 235
18 Poisson-Type Summation Formulas over Euclidean Spaces 237
18.1 Integral Means for Iterated Euler–Green Functions . . . . . . 237
18.2 Euler-TypeSummationFormulaoverIncreasingBallsInvolving
Euler–Green Functions . . . . . . . . . . . . . . . . . . . . . 241
18.3 Spherically-Reflected Convergence Criteria (Basic Differentia-
bility Order) . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
18.4 Poisson-Type Summation Formula (Heuristic Approach) . . 246
18.5 Euler-TypeSummationFormulaoverIncreasingBallsInvolving
Iterated Euler–Green Functions . . . . . . . . . . . . . . . . 247
18.6 Spherically-ReflectedConvergenceCriteria(HigherDifferentia-
bility Orders) . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
18.7 Poisson-Type Summation Formula (Rigorous Approach) . . 253
18.8 Hardy–Landau-Type Identities (Spherical Harmonic Weights) 255
IX Multivariate Shannon-Type Sampling
Formulas over Euclidean Spaces 261
19 Shannon-Type Sampling Based on Poisson-Type Summation
Formulas over Euclidean Spaces 263
19.1 Functional Values–Based Shannon-Type Sampling . . . . . . 264
19.2 Paley–Wiener Reproducing Kernel Structure . . . . . . . . . 267
19.3 Fourier Transformed Values–Based Shannon-Type Sampling 268
19.4 Shannon-Type Sampling Involving Dilated Fundamental Cells 271
19.5 Bivariate Locally-Supported Sampling Functions . . . . . . . 272
19.6 From Gaussian to Ordinary Non-Bandlimited Shannon-Type
Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
X Conclusion 279
20 Trends, Progress, and Perspectives 281
20.1 Trendsetting Extensions of Shannon Sampling . . . . . . . . 281
20.2 Methodological Progress in Sampling . . . . . . . . . . . . . 282
20.3 Bridging Role of Sampling in Recovery Problems . . . . . . . 284
20.4 SampTA Conference Series . . . . . . . . . . . . . . . . . . . 285
Bibliography 287
Index 301