Generalized discrete orbit function transforms of affine Weyl groups

TL;DR

This paper develops generalized discrete orbit function transforms based on affine Weyl groups, providing a unified framework that includes classical cosine and sine transforms as special cases.

Contribution

It introduces a new class of discrete orbit function transforms associated with affine Weyl groups, including classification of admissible shifts and construction of orthogonal sampling grids.

Findings

Complete sets of discretely orthogonal orbit functions are established.

Discrete Fourier transforms are formulated based on these orbit functions.

Classical discrete cosine and sine transforms are shown as special cases.

Abstract

The affine Weyl groups with their corresponding four types of orbit functions are considered. Two independent admissible shifts, which preserve the symmetries of the weight and the dual weight lattices, are classified. Finite subsets of the shifted weight and the shifted dual weight lattices, which serve as a sampling grid and a set of labels of the orbit functions, respectively, are introduced. The complete sets of discretely orthogonal orbit functions over the sampling grids are found and the corresponding discrete Fourier transforms are formulated. The eight standard one-dimensional discrete cosine and sine transforms form special cases of the presented transforms.