The discrete SU(3)SU(3) transform and its continuous extension for triangular lattices

We develop a new method for the discrete Fourier-type transform of multi-dimensional grid functions, which is based on orbit functions of compact Lie groups of different symmetries. Here we present an implementation of this transform, abbreviated here as DOFT (for Discrete Orbit Function Transform  ), for a 2-dimensional discrete function {fk,m}{fk,m} produced by sampling of a continuous function f(z)f(z) with z∈R2z∈R2 on the points zk,mzk,m of an equilateral triangular grid FN with NN equal subintervals along each of its edges. The DOFT for such a triangular grid corresponds to implementation of the case of Lie group SU(3)SU(3). We present the mathematical details for realization of this case, and show that the method provides an exact solution to the problem of the discrete Fourier-type transform for such symmetry grids. In this paper we also continue development of the approach of generalization of a discrete inverse transform to the form of a continuous trigonometric polynomial. We describe and exemplify the properties of such a continuous extension   of the DOFT (abbreviated as CEDOFT) for the SU(3)SU(3) group, which proves an effective tool for interpolation of the discrete function onto all points of the triangular region F. Like in the case of the CEDOFT on SU(2)SU(2) studied in detail earlier, the CEDOFT on SU(3)SU(3) has very good properties of convergence with the increase of NN. It also shows localization and differentiation properties, which can be useful for a number of practical applications.