Discrete Fourier analysis with lattices on planar domains

TL;DR

This paper develops a discrete Fourier analysis framework using translation lattices on planar domains, enabling new cubature and interpolation results with trigonometric and algebraic polynomials.

Contribution

It introduces a novel approach to discrete Fourier analysis with two lattices, expanding the understanding of tiling and polynomial approximation on planar domains.

Findings

New cubature formulas for planar domains

Enhanced interpolation methods using lattice-based Fourier analysis

Identification of lattice configurations that tile the plane

Abstract

A discrete Fourier analysis associated with translation lattices is developed recently by the authors. It permits two lattices, one determining the integral domain and the other determining the family of exponential functions. Possible choices of lattices are discussed in the case of lattices that tile $\RR^2$ and several new results on cubature and interpolation by trigonometric, as well as algebraic, polynomials are obtained.