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# Metaharmonic Lattice Point Theory

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## Book Description

**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(s)

### Biography

**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.