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

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. Keywordsalgebraic signal processing theory • Christoffel-Darboux formula • discrete cosine transform • fast Fourier transform • Gauss-Jacobi procedure • multivariate Chebyshev polynomials • orthogonal polynomials • representation theory of algebras • root systems • weight lattices Mathematics Subject Classification (2010) 65T50 • 15A23 • 33F99 • 68R01 • 16G99This work is partially supported by the European Regional Development Fund (ERDF).