Metaharmonic Lattice Point Theory

Metaharmonic Lattice Point Theory

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Features

  • Presents multi-dimensional techniques for periodization
  • Focuses on geomathematically based/oriented tools and procedures
  • Describes weighted lattice point and ball numbers in georelevant "potato-like" regions
  • Discusses radial and angular non-uniform lattice point distribution

Summary

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.

Table of Contents

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

Author Bio(s)

Willi Freeden is the head of the Geomathematics Group in the Department of Mathematics at the University of Kaiserslautern, where he has been vice president for research and technology. Dr. Freeden is also editor-in-chief of the International Journal on Geomathematics. His research interests include special functions of mathematical geophysics, partial differential equations, constructive approximation, numerical methods and scientific computing, and inverse problems in geophysics, geodesy, and satellite technology.

 
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