Algebra of Orthogonal Series - Pankratov, A. N.; Tetuev, R. K. (Pattern recognition and image analysis, 2023)
An algebra of orthogonal series has been developed for the operations of multiplication and division, differentiation and integration of signals represented by series of classical orthogonal polynomials and functions. It is shown that the considered linear transformations for classical orthogonal polynomials and functions are subject to recurrence relations of a special form, which makes it possible to perform these transformations over series in the space of expansion coefficients. A theorem on the condition for the existence of a recurrence relation for the inverse transformation is proven. A general scheme of an algorithm for calculating the coefficients of the resulting series from the coefficients of the original series through the coefficients of recurrent relations is proposed. It has been proven that the linear transformation of a series of length is executed by a linear complexity algorithm . Formulas for the Chebyshev, Jacobi, Laguerre, and Hermite polynomials are given.