Metaharmonic lattice point theory

 Content: 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
 Abstract:             Metaharmonic Lattice Point Theory covers interrelated methods and tools of spherically oriented geomathematics and periodically reflected analytic number theory. The book establishes multi-dimensional Euler and Poisson summation formulas corresponding to elliptic operators for the adaptive determination and calculation of formulas and identities of weighted lattice point numbers, in particular the non-uniform distribution of lattice points. The author explains how to obtain multi-dimensional generalizations of the Euler summation formula by interpreting classical Bernoulli polynomials as Green\'s functions and linking them to Zeta and Theta functions. To generate multi-dimensional Euler summation formulas on arbitrary lattices, the Helmholtz wave equation must be converted into an associated integral equation using Green\'s functions as bridging tools. After doing this, the weighted sums of functional values for a prescribed system of lattice points can be compared with the corresponding integral over the function. Exploring special function systems of Laplace and Helmholtz equations, this book focuses on the analytic theory of numbers in Euclidean spaces based on methods and procedures of mathematical physics. It shows how these fundamental techniques are used in geomathematical research areas, including gravitation, magnetics, and geothermal