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Fast Decompositions by Gegenbauer Family Orthogonal Polynomials

Anry Nersessian
Organization: National Academy of Sciences of Armenia
Department: Institute of Mathematics
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The efficient expansions by classical orthogonal polynomials is a well known and hard solving problem for applied mathematics. At first sight there is no problem as Mathematica has a package for the fast generation of all known classical orthogonal polynomials and has a powerful package for numerical integration. However when a user makes an expansion of a smooth function the answers come very slowly and with significant errors. For example, it happens when it is necessary to have more than 20 terms in Gegenbauer expansion in a one-dimensional case (see [1]). The main reason of this phenomenon is high oscillation of all known functional orthogonal systems if the corresponding index is rather big. This package is based on the original papers of the author and contains an absolutely new approach for calculation of expansions by orthogonal polynomials.

*Mathematics > Algebra > Polynomials

orthogonal polynomial, Gegenbauer
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GegenbauerPolynomials.nb (3.1 MB) - Mathematica notebook