Discrete and continuous cosine transform generalized to Lie groups SU(3) and G(2)

Abstract

In the paper we complete the development and description of the four variants of two-dimensional generalization of the cosine transform started in [Patera and Zaratsyan, J. Math Phys. 46, 053514 (2005)]. Each variant is based on a compact semisimple Lie group G of rank 2. Here, the groups are SU(3) and G(2). The cosines are generalized as the corresponding C-functions of the Lie group. A C-function is the contribution to an irreducible character from one orbit of the appropriate Weyl group. An explicit description is provided for expansions of functions given on the fundamental region F of the two compact simple Lie groups into series of C-functions. The fundamental region F is an equilateral triangle for SU(3) and half of such a triangle for G(2). Expansion coefficients are calculated using orthogonality of C-functions on F. Discrete expansions are set up on a grid FM⊂F. The grid is defined group theoretically for all positive integers M. It consists of points in F that represent conjugacy classes of elements of the finite maximal Abelian subgroup of G generated by its elements of order M. The C-functions are orthogonal on such a grid; hence, coefficients of discrete expansions are calculated independently of the continuous expansions. Processing digital data, sampled on triangular lattices, is the motivating application here.