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.
Highlights
The goal of this article is to develop discrete transforms of the multivariate symmetric and antisymmetric sine functions [1] together with the related Fourier interpolation and Chebyshev polynomial methods
The cubature formulas of multivariate generalizations of the classical Chebyshev polynomials of the second and fourth kind are induced by the discrete transforms
The multivariatesymmetric trigonometric functions lead to the generalizations of the Chebyshev polynomials and the discrete transforms produce effective interpolation methods along with the related cubature integration formulas
Summary
The goal of this article is to develop discrete transforms of the multivariate symmetric and antisymmetric sine functions [1] together with the related Fourier interpolation and Chebyshev polynomial methods. The cubature formulas of multivariate generalizations of the classical Chebyshev polynomials of the second and fourth kind are induced by the discrete transforms. The multivariate (anti)symmetric trigonometric functions lead to the generalizations of the Chebyshev polynomials and the discrete transforms produce effective interpolation methods along with the related cubature integration formulas. The 2D and 3D (anti)symmetric sine functions as eigenfunctions of the discretized Laplace operator potentially represent solutions to lattice vibration models in solid state physics as well as foundation for description of the corresponding models in quantum field theory [23] Boundary conditions of these models are determined by the boundary behavior of the multivariate discrete transforms and the spectral analysis provided by the developed transforms contributes to explicit solutions of the time evolution of the mechanical models.
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