Lattice point identities and Shannon-type sampling

Lattice point identities and Shannon-type sampling

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 varian...

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Bibliographic Details
Main Author: Freeden, W. (Willi)
Other Authors: Nashed, M. Zuhair
Format: Electronic eBook
Language:English
Published: London : CRC Press LLC, 2019.
Series:Chapman and Hall/CRC Monographs and Research Notes in Mathematics Ser.
Subjects:
Sampling (Statistics)
Lattice theory.
Online Access:Taylor & Francis
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Table of Contents:
  • 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)

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