Lattice Point Identities and Shannon-Type Sampling Lattice Point Identities and Shannon-Type Sampling Willi Freeden M. Zuhair Nashed CRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 BocaRaton,FL33487-2742 (cid:13)c 2020byTaylor&FrancisGroup,LLC CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness NoclaimtooriginalU.S.Governmentworks Printedonacid-freepaper InternationalStandardBookNumber-13:978-0-367-37563-8(Hardback) Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Rea- sonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the conse- quences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 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