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This book leads the reader through a research excursion, beginning from the Gaussian circle problem of the early nineteenth century, via the classical Hardy-Landau lattice point identity and the Hardy conjecture of the first half of the twentieth century, and the Shannon sampling theorem (its variants, generalizations and the fascinating stories about the cardinal series) of the second half of the twentieth century. The authors demonstrate how all these facets have resulted in new multivariate extensions of lattice point identities and Shannon-type sampling procedures of high practical applicability, thereby also providing a general reproducing kernel Hilbert space structure of an associated Paley-Wiener theory over (potato-like) bounded regions (cf. the cover illustration of the geoid), as well as the whole Euclidean space.

Description based upon print version of record.

Cover; Half Title; Series Page; Title Page; Copyright Page; Contents; Preface; Authors; Acknowledgments; Part I: Central Theme; 1. From Lattice Point to Shannon-Type Sampling Identities; 1.1 Classical Framework of Shannon Sampling; 1.2 Transition From Shannon to Shannon-Type Sampling; 1.3 Novel Framework of Shannon-Type Sampling; 2. Obligations, Ingredients, Achievements, and Innovations; 2.1 Obligations and Ingredients; 2.2 Achievements and Innovative Results; 2.3 Methods and Tools; 3. Layout; 3.1 Structural Organisation; 3.2 Relationship to Other Monographs

Part II: Univariate Poisson-Type Summation Formulas and Shannon-Type Sampling4. Euler/Poisson-Type Summation Formulas and Shannon-Type Sampling; 4.1 Classical Euler Summation Formula; 4.2 Variants of the Euler Summation Formula; 4.3 Poisson-Type Summation Formula over Finite Intervals; 4.4 Shannon Sampling Based on the Poisson Summation-Type Formula; 4.5 Shannon-Type Sampling Based on Poisson Summation-Type Formulas; 4.6 Fourier Transformed Values-Based Shannon-Type Sampling (Finite Intervals); 4.7 Functional Values-Based Shannon-Type Sampling (Finite Intervals)

4.8 Paley-Wiener Reproducing Kernel Hilbert Spaces4.9 Poisson-Type Summation Formula over the Euclidean Space; 4.10 Functional Values-Based Shannon-Type Sampling (Euclidean Space); 4.11 Fourier Transformed Values-Based Shannon-Type Sampling (Euclidean Space); Part III: Preparatory Material for Multivariate Lattice Point Summation and Shannon-Type Sampling; 5. Preparatory Tools of Vector Analysis; 5.1 Cartesian Notation and Settings; 5.2 Spherical Notation and Settings; 5.3 Regular Regions and Integral Theorems; 6. Preparatory Tools of the Theory of Special Functions

6.1 Homogeneous Harmonic Polynomials6.2 Bessel Functions; 6.3 Asymptotic Expansions; 7. Preparatory Tools of Lattice Point Theory; 7.1 Lattices in Euclidean Spaces; 7.2 Figure Lattices in Euclidean Spaces; 7.3 Basic Results of the Geometry of Numbers; 7.4 Lattice Points Inside Spheres; 8. Preparatory Tools of Fourier Analysis; 8.1 Stationary Point Asymptotics; 8.2 Periodic Polynomials and Fourier Expansions; 8.3 Fourier Transform over Euclidean Spaces; 8.4 Periodization and Classical Poisson Summation Formula; 8.5 Gauss-Weierstrass Transform over Euclidean Spaces

8.6 Hankel Transform and Discontinuous IntegralsPart IV: Multivariate Euler-Type Summation Formulas over Regular Regions; 9. Euler-Green Function and Euler-Type Summation Formula; 9.1 Euler-Green Function; 9.2 Euler-Type Summation Formulas over Regular Regions Based on Euler-Green Functions; 9.3 Iterated Euler-Green Function; 9.4 Euler-Type Summation Formulas over Regular Regions Based on Iterated Euler-Green Functions; Part V: Bivariate Lattice Point/Ball Summation and Shannon-Type Sampling; 10. Hardy-Landau-Type Lattice Point Identities (Constant Weight)

10.1 Integral Mean Asymptotics for the Euler-Green Function

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