Fourier transform over semi-simple algebras and harmonic analysis for probabilistic algorithms

However, these powers are often hard to calculate in a nice closed form. For this reason we shall consider their Fourier transforms in some suitable context. The idea being that intricate computation of products in the group algebra are replaced by simple point wise products. More precisely, we shall derive new extended versions of a formula of Garsia (see (1) below) and of similar formulas obtained in [4], as well as generalizations of those. These formulas can be considered to give, in explicit form, a Fourier transform such as defined below. Let A be a semisimple algebra, and let B = v1, v2, . . . , vn be some fixed (linear) basis for the subalgebra B, of A, spanned by the complete set of primitive idempotents e1, e2, . . . , en of A. Recall that these idempotents are such that