Discrete transforms and orthogonal polynomials of (anti)symmetric multivariate cosine functions

TL;DR

This paper extends discrete cosine transforms to symmetric and antisymmetric multivariate cases, introduces new orthogonal polynomial families, and derives related cubature formulas and interpolation methods.

Contribution

It generalizes discrete cosine transforms to multivariate symmetric and antisymmetric cases, introducing new orthogonal polynomial families and deriving associated cubature and interpolation formulas.

Findings

Introduces four families of orthogonal Chebyshev-like polynomials.

Develops cubature formulas based on these polynomial families.

Provides explicit 3D interpolation formulas and polynomial forms.

Abstract

The discrete cosine transforms of types V--VIII are generalized to the antisymmetric and symmetric multivariate discrete cosine transforms. Four families of discretely and continuously orthogonal Chebyshev-like polynomials corresponding to the antisymmetric and symmetric generalizations of cosine functions are introduced. Each family forms an orthogonal basis of the space of all polynomials with respect to some weighted integral. Cubature formulas, which correspond to these families of polynomials and which stem from the developed discrete cosine transforms, are derived. Examples of three-dimensional interpolation formulas and three-dimensional explicit forms of the polynomials are presented.