Inner product and Gegenbauer polynomials in Sobolev space

Abstract

In this paper we consider the system of functions G_(r,n)^α (x) (r∈N,n=0,1,…) which is orthogonal with respect to the Sobolev-type inner product on (-1,1) and generated by orthogonal Gegenbauer polynomials. The main goal of this work is to study some properties related to the system {φ_(k,r) (x)}_(k≥0) of the functions generated by the orthogonal system {G_(r,n)^α (x)} of Gegenbauer functions. We study the conditions on a function f(x) given in a generalized Gegenbauer orthogonal system for it to be expandable into a generalized mixed Fourier series of the form f(x)~∑_(k=0)^(r-1)▒〖f^((k) ) (-1) (x+1)^k/k!+∑_(k=r)^∞▒〖G_(r,k)^α (f) 〗〗 φ_(r,k)^α (x), as well as the convergence of this Fourier series. The second result of this paper is the proof of a recurrence formula for the system {φ_(k,r) (x)}_(k≥0). We also discuss the asymptotic properties of these functions, and this represents the latter result of our contribution.