Hermite functions and inner product in Sobolev space

Abstract

Let us consider the orthogonal Hermite system 〖{φ_2n (x)}〗_(n≥0) of even index defined on (-∞,∞), where φ_2n (x)=e^(-x^2/2)/(√((2n)!) π^(1/4) 2^n ) H_2n (x), by H_2n (x) we denote a Hermite polynomial of degree 2n. In this paper, we consider a generalized system {ψ_(r,2n) (x)} with r>0, n≥0 which is orthogonal with respect to the Sobolev type inner product on (-∞,∞), i.e. 〈f,g〉=lim┬(t→-∞)⁡∑_(k=0)^(r-1)▒〖f^((k) ) (t) g^((k) ) (t)+∫_(-∞)^∞▒〖f^((r) ) (x) g^((r) ) (x)ρ(x)dx〗〗 with ρ(x)=e^(-x^2 ), and generated by 〖{φ_2n (x)}〗_(n≥0). The main goal of this work is to study some properties related to the system 〖{ψ_(r,2n) (x)}〗_(n≥0), ψ_(r,n) (x)=〖(x-a)〗^n/n!,n=0,1,2,…,r-1, ψ_(r,r+n) (x)=1/(r-1)! ∫_a^b▒〖(x-t)^(r-1) φ_n (t)dt,〗 n=0,1,2,… . We study the conditions on a function f(x), given in a generalized Hermite orthogonal system, for it to be expandable into a generalized mixed Fourier series as well as the convergence of this Fourier series. The second result of the paper is the proof of a recurrent formula for the system 〖{ψ_(r,2n) (x)}〗_(n≥0). We also discuss the asymptotic properties of these functions, and this concludes our contribution.