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Alternating multivariate trigonometric functions and corresponding Fourier transforms. (English) Zbl 1183.33027

Summary: We define and study multivariate sine and cosine functions, symmetric with respect to the alternating group \(A_n\), which is a subgroup of the permutation (symmetric) group \(S_n\). These functions are eigenfunctions of the Laplace operator. They determine Fourier-type transforms. There exist three types of such transforms: expansions into corresponding sine-Fourier and cosine-Fourier series, integral sine-Fourier and cosine-Fourier transforms, and multivariate finite sine and cosine transforms. In all these transforms, alternating multivariate sine and cosine functions are used as a kernel.

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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