Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization

Abstract

A versatile method is described for the practical computation of the exact discrete Fourier transforms (DFT), both the direct and the inverse ones, of a continuous function g given by its values gj at the points of a uniform grid FN generated by conjugacy classes of elements of finite adjoint order N in the fundamental region F of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when F is reduced to a one-dimensional segment, and for SU(2)×SU(2)×⋯×SU(2) in multidimensional cases. This simplest case turns out to be a version of the discrete cosine transform (DCT). Implementations, abbreviated as DGT for Discrete Group Transform, based on simple Lie groups of higher ranks, are to be considered separately. DCT is often taken to be simply a specific type of the standard DFT. Here we show that the DCT is very different from the standard DFT when the properties of the continuous extensions of the two inverse discrete transforms are studied. The following properties of the continuous extension of DCT (called CEDCT) from the discrete tj∈FN to all t∈F are proven and exemplified. Like the standard DFT, the DCT also returns the exact values of {gj} on the N+1 points of the grid. However, unlike the continuous extension of the standard DFT: (a) The CEDCT function fN(t) closely approximates g(t) between the points of the grid as well; (b) for increasing N, the derivative of fN(t) converges to the derivative of g(t); (c) for CEDCT the principle of locality is valid. In this article we also use the continuous extension of the two-dimensional (2D) DCT, SU(2)×SU(2), to illustrate its potential for interpolation as well as for the data compression of 2D images.