Approximation of Hermite polynomials by biorthognal systerms

Abstract

In this paper, the structure to a family of Appell sequences that approximate to Hermite polynomials is investigated by the functional ø which approximates to Gaussian function to construct the biothogonal systems between the sequences and the derivatives of ø. Therefore, the asymptotic relations between several orthogonal polynomials and combinatoric polynomials are derived from the biothogonal systems. Especially, the Appell sequences generated by the uniform B-splines of order <italic>N</italic> are Bernoulli polynomials of order <italic>N</italic> which indicate the biorthogonal relationship between Bernoulli polynomials and the derivatives of B-splines. Therefore, the standardized Bernoulli polynomials approximate to Hermite polynomials. The asymptotic properties of standardized Euler polynomials to Hermite polynomials are derived by the biothognal systems generated by the binomial distribution and Euler polynomials. The judging theorem of the approximation to Hermite polynomials by a sequence of functions and the necessary and sufficient condition of the generating functions to the Appell sequence which satisfies the scaling equations are also discussed. The asymptotic representations of generalized Buchholz, Laguerre and Ultraspherical (Gegenbauer) polynomials to Hermite polynomials are proved by the theorems which in turn verify the Askey scheme of hypergeometric orthogonal polynomials.