# On E-Discretization of Tori of Compact Simple Lie Groups: II

## TL;DR

This paper develops ten types of discrete Fourier transforms based on Weyl orbit functions, extending classical transforms and establishing their orthogonality, symmetry, and explicit counting formulas for applications in harmonic analysis on Lie groups.

## Contribution

It introduces ten new discrete Fourier transform types derived from Weyl orbit functions, including real-valued Hartley transforms, with detailed orthogonality and counting formulas.

## Key findings

- Ten types of discrete Fourier transforms developed
- Explicit formulas for the number of transform points
- Introduction of real-valued Hartley orbit functions

## Abstract

Ten types of discrete Fourier transforms of Weyl orbit functions are developed. Generalizing one-dimensional cosine, sine and exponential, each type of the Weyl orbit function represents an exponential symmetrized with respect to a subgroup of the Weyl group. Fundamental domains of even affine and dual even affine Weyl groups, governing the argument and label symmetries of the even orbit functions, are determined. The discrete orthogonality relations are formulated on finite sets of points from the refinements of the dual weight lattices. Explicit counting formulas for the number of points of the discrete transforms are deduced. Real-valued Hartley orbit functions are introduced and all ten types of the corresponding discrete Hartley transforms are detailed.