Two dimensional symmetric and antisymmetric generalizations of sine functions

TL;DR

This paper introduces symmetric and antisymmetric 2D sine functions, exploring their orthogonality, product decompositions, and applications for Fourier-like expansions and interpolation of digital data on 2D lattices.

Contribution

It develops a formalism for 2D sine functions with symmetry properties, enabling Fourier-like analysis and interpolation of digital data on lattices.

Findings

Functions are orthogonal over finite regions.

Functions are discretely orthogonal on lattices.

Products of functions can be decomposed into sums.

Abstract

Properties of 2-dimensional generalizations of sine functions that are symmetric or antisymmetric with respect to permutation of their two variables are described. It is shown that the functions are orthogonal when integrated over a finite region $F$ of the real Euclidean space, and that they are discretely orthogonal when summed up over a lattice of any density in $F$. Decomposability of the products of functions into their sums is shown by explicitly decomposing products of all types. The formalism is set up for Fourier-like expansions of digital data over 2-dimensional lattices in $F$. Continuous interpolation of digital data is studied.