Discrete Transforms and Orthogonal Polynomials of (Anti)symmetric Multivariate Sine Functions

Abstract

Sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated. The four generated classes of the corresponding orthogonal polynomials generalize the formation of the Chebyshev polynomials of the second and fourth kinds. Continuous orthogonality relations of the polynomials together with the inherent weight functions are deduced. Sixteen cubature rules, including the four Gaussian, are produced by the related discrete transforms. For the three-dimensional case, interpolation tests, unitary transform matrices and recursive algorithms for calculation of the polynomials are presented.

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

“…In the fourth paper [ 21 ], authored by Adam Brus, Jiří Hrivnák, and Lenka Motlochová, sixteen types of the discrete multivariate transforms, induced by the multivariate antisymmetric and symmetric sine functions, are explicitly developed. Provided by the discrete transforms, inherent interpolation methods are formulated.…”

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

“…The established correspondence [5] between the (anti)symmetric trigonometric functions and Weyl orbit functions induced by the crystallographic root systems A 1 and C n [6,7] is utilized to interlace the kernels, point and label sets of the discrete transforms. The exact connections among the types of the discrete transforms allow for comparison and migration of the associated multivariate Fourier and Chebyshev methods [8][9][10].…”

“…There exist eight symmetric multivariate discrete cosine transforms (SMDCTs) of types I-VIII and eight antisymmetric multivariate discrete cosine transforms (AMDCTs) of types I-VIII [9]. In addition to the generalized discrete cosine transforms, eight symmetric multivariate discrete sine transforms (SMDSTs) of types I-VIII and eight antisymmetric multivariate discrete sine transforms (AMDSTs) of types I-VIII emerge from the eight standard discrete sine transforms [1,10,12]. Moreover, the antisymmetric trigonometric transforms constitute special cases of the Fourier transforms evolved from the generalized Schur polynomials [13].…”

“…The symmetry properties of the (anti)symmetric trigonometric functions restrict both points and labels of the corresponding discrete transforms to specifically constructed fundamental domains. The unique positioning of the transform nodes and labels relative to their respective fundamental domains reflects the symmetries and boundary behavior of each type of the (anti)symmetric trigonometric function [9,10]. Resolving the exact isomorphisms between the symmetry groups of the (anti)symmetric trigonometric functions and the affine Weyl groups related to the series of the crystallographic root systems A 1 and C n (C n series) leads to the explicit direct link between the (anti)symmetric trigonometric functions and Weyl orbit functions [5].…”

“…It appears that only after taking into account recent Fourier-Weyl transforms with the rescaled shifted dual root lattices point sets and the shifted weight lattices label sets [3,4], the entire family of 32 multivariate discrete (anti)symmetric trigonometric transforms permits embedding into the Fourier-Weyl formalism of the crystallographic root systems C n . Since both (anti)symmetric trigonometric and Fourier-Weyl transforms attain uniform characterization by their complementary unitary transform matrices [4,10], coupling together the ordered label and point sets as well as the weight and normalization functions produces the exact one-to-one correspondence between the unitary matrices of the induced discrete transforms.…”