Abstract
In the article, the theory of the Fourier series on the orthogonal multidimensional-matrix polynomials is developed. The known results from the theory of the orthogonal polynomials of the vector variable and the Fourier series are given and the new results are presented. In particular, the known results of the Fourier series are extended to the case of the multidimensional-matrix functions, what allows us to solve more general approximation problems. The general case of the approximation of the multidimensional-matrix function of the vector argument by the Fourier series on the orthogonal multidimensional-matrix polynomials is realized programmatically as the program function and its efficiency is confirmed. The analytical expressions for the coefficients of the second degree orthogonal polynomials and Fourier series for the potential studies are obtained.
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