Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions

TL;DR

This paper explores symmetric and antisymmetric two-variable special functions, analyzing their properties and potential for Fourier-based digital data interpolation on square grids.

Contribution

It introduces and studies four families of symmetric and antisymmetric special functions of two variables, detailing their properties for Fourier expansion applications.

Findings

Functions are suitable for Fourier expansions of digital data.

Interpolation quality is analyzed and compared.

Functions are applicable for data sampled on square grids.

Abstract

Properties of the four families of recently introduced special functions of two real variables, denoted here by $E^\pm$, and $\cos^\pm$, are studied. The superscripts $^+$ and $^-$ refer to the symmetric and antisymmetric functions respectively. The functions are considered in all details required for their exploitation in Fourier expansions of digital data, sampled on square grids of any density and for general position of the grid in the real plane relative to the lattice defined by the underlying group theory. Quality of continuous interpolation, resulting from the discrete expansions, is studied, exemplified and compared for some model functions.