Metaharmonic Lattice Point Theory

ISBN: 9781439861844 出版年：2011 页码：467 Freeden, Willi CRC Press

Introduction Historical Aspects Preparatory Ideas and Concepts Tasks and Perspectives Basic Notation Cartesian Nomenclature Regular Regions Spherical Nomenclature Radial and Angular Functions One-Dimensional Auxiliary Material Gamma Function and Its Properties Riemann-Lebesgue Limits Fourier Boundary and Stationary Point Asymptotics Abel-Poisson and Gauss-Weierstrass Limits One-Dimensional Euler and Poisson Summation Formulas Lattice Function Euler Summation Formula for the Laplace Operator Riemann Zeta Function and Lattice Function Poisson Summation Formula for the Laplace Operator Euler Summation Formula for Helmholtz Operators Poisson Summation Formula for Helmholtz Operators Preparatory Tools of Analytic Theory of Numbers Lattices in Euclidean Spaces Basic Results of the Geometry of Numbers Lattice Points Inside Circles Lattice Points on Circles Lattice Points Inside Spheres Lattice Points on Spheres Preparatory Tools of Mathematical Physics Integral Theorems for the Laplace Operator Integral Theorems for the Laplace-Beltrami Operator Tools Involving the Laplace Operator Radial and Angular Decomposition of Harmonics Integral Theorems for the Helmholtz-Beltrami Operator Radial and Angular Decomposition of Metaharmonics Tools Involving Helmholtz Operators Preparatory Tools of Fourier Analysis Periodical Polynomials and Fourier Expansions Classical Fourier Transform Poisson Summation and Periodization Gauss-Weierstrass and Abel-Poisson Transforms Hankel Transform and Discontinuous Integrals Lattice Function for the Iterated Helmholtz Operator Lattice Function for the Helmholtz Operator Lattice Function for the Iterated Helmholtz Operator Lattice Function in Terms of Circular Harmonics Lattice Function in Terms of Spherical Harmonics Euler Summation on Regular Regions Euler Summation Formula for the Iterated Laplace Operator Lattice Point Discrepancy Involving the Laplace Operator Zeta Function and Lattice Function Euler Summation Formulas for Iterated Helmholtz Operators Lattice Point Discrepancy Involving the Helmholtz Operator Lattice Point Summation Integral Asymptotics for (Iterated) Lattice Functions Convergence Criteria and Theorems Lattice Point-Generated Poisson Summation Formula Classical Two-Dimensional Hardy-Landau Identity Multi-Dimensional Hardy-Landau Identities Lattice Ball Summation Lattice Ball-Generated Euler Summation Formulas Lattice Ball Discrepancy Involving the Laplacian Convergence Criteria and Theorems Lattice Ball-Generated Poisson Summation Formula Multi-Dimensional Hardy-Landau Identities Poisson Summation on Regular Regions Theta Function and Gauss-Weierstrass Summability Convergence Criteria for the Poisson Series Generalized Parseval Identity Minkowski's Lattice Point Theorem Poisson Summation on Planar Regular Regions Fourier Inversion Formula Weighted Two-Dimensional Lattice Point Identities Weighted Two-Dimensional Lattice Ball Identities Planar Distribution of Lattice Points Qualitative Hardy-Landau Induced Geometric Interpretation Constant Weight Discrepancy Almost Periodicity of the Constant Weight Discrepancy Angular Weight Discrepancy Almost Periodicity of the Angular Weight Discrepancy Radial and Angular Weights Non-Uniform Distribution of Lattice Points Quantitative Step Function Oriented Geometric Interpretation Conclusions Summary Outlook Bibliography Index