# FFT and orthogonal discrete transform on weight lattices of semi-simple Lie groups

## TL;DR

This paper introduces two algebraic approaches to develop fast Fourier transform algorithms on weight lattices of semi-simple Lie groups, extending classical methods to multivariate orthogonal polynomials.

## Contribution

It presents novel algebraic methods for multivariate Fourier transforms on Lie group lattices, including module induction and polynomial decomposition techniques.

## Key findings

- Developed two algebraic algorithms for multivariate Fourier transforms.
- Extended Gauss-Jacobi procedure to multivariate orthogonal polynomials.
- Provided a scheme for fast transforms based on multivariate Chebyshev polynomials.

## Abstract

We give two algebro-geometric inspired approaches to fast algorithms for Fourier transforms in algebraic signal processing theory based on polynomial algebras in several variables. One is based on module induction and one is based on a decomposition property of certain polynomials. The Gauss-Jacobi procedure for the derivation of orthogonal transforms is extended to the multivariate setting. This extension relies on a multivariate Christoffel-Darboux formula for orthogonal polynomials in several variables. As a set of application examples a general scheme for the derivation of fast transforms of weight lattices based on multivariate Chebyshev polynomials is derived. A special case of such transforms is considered, where one can apply the Gauss-Jacobi procedure.