Discrete and continuous cosine transform generalized to Lie groups SU(2)×SU(2) and O(5)

Abstract

We develop and describe continuous and discrete transforms of class functions on compact semisimple Lie group G as their expansions into series of uncommon special functions, called here C-functions in recognition of the fact that the functions generalize cosine to any dimension n<∞. A uniform discretization of the problem on lattices of any density is described. Continuous and discrete orthogonality of C-functions is shown. Discrete transform is known in the case n=1 as the cosine transform. Continuous extension of the discrete transform is described. In general, C-functions are the contributions to irreducible characters from just one orbit of the Weyl group of G. Their products are fully decomposable to the sums of C-functions, so are the reductions to subgroups of the Lie group. They are eigenfunctions of Laplace operator, satisfying Neumann conditions at the boundary of the fundamental region of G, etc. A ready-to-use presentation is made of two of the four variants of the two-dimensional transforms. Both variants have in common exploitation of square lattices for the discrete version of the transforms. They are based on the compact Lie groups SU(2)×SU(2) and O(5), or, equivalently, Sp(4). Remaining two groups, SU(3) and G(2), involve triangular lattices. They are considered separately. Processing digital data, sampled on square lattices, is our motivating application.