Central Splitting of A2 Discrete Fourier–Weyl Transforms

Abstract

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

Connecting (Anti)Symmetric Trigonometric Transforms to Dual-Root Lattice Fourier–Weyl Transforms

“…The successful interpolation tests for the 2D and 3D (anti)symmetric trigonometric functions [2,9,10,12] indicate matching interpolation behavior of the corresponding Fourier-Weyl transforms as well as relevance of both types of transforms in data processing methods [20]. In particular, the vast pool of recursive algorithms for fast computation of the (multivariate) trigonometric transforms [11] becomes linked to the central splitting of the discrete Fourier-Weyl transforms [21,22] and vice versa. Moreover, the cubature rules [23] of the multivariate Chebyshev polynomials, that are obtained from the (anti)symmetric trigonometric functions [9,10] and associated with the Jacobi polynomials [5,24,25], are further intertwined with the Lie theoretical approach [8,26,27].…”