Hermite functions and inner product in Sobolev space

Let us consider the orthogonal Hermite system $\{{{\phi }_{2}n\left(x\right)\}}_{n\ge 0}$ of even index defined on $(-\infty ,\infty )$, where ${\phi }_{2}n\left(x\right)=\frac{e^{-{\displaystyle \frac{x^2}{2}}}}{\surd \left(2n\right)!{\pi }^{\frac{1}{4}}{2}^n)}{H}_{2}n\left(x\right),$

by $H_{2}n\left(x\right)$ we denote a Hermite polynomial of degree $2n$. In this paper, we consider a generalized system $\left\{{\psi }_{r,2n}\right(x\left)\right\}$ with $r>0$, $n\ge 0$ which is orthogonal with respect to the Sobolev type inner product on $(-\infty ,\infty )$, i.e. $⟨f,g⟩=\underset{(t\to -\infty )}{lim}\sum _{k=0}^{r-1}{f}^{\left(k\right)}\left(t\right)g^{\left(k\right)}\left(t\right)+\underset{-\infty }{\overset{{}^{\infty }}{\int }}{f}^{{}^{\left(r\right)}}\left(x\right)g^{\left(r\right)}\left(x\right)\rho \left(x\right)dx$

with $\rho \left(x\right)=e^{(}-x^2),$ and generated by ${\left\{{\phi }_{2}n\left(x\right)\right\}}_{n\ge 0}$. The main goal of this work is to study some properties related to the system ${\left\{{\psi }_{r,2n}\left(x\right)\right\}}_{n\ge 0}$, ${\psi }_{r,n}\left(x\right)=\frac{{(x-a)}^n}{n!}, n=0,1,2,\dots , r-1,$  ${\psi }_{r,r+n}\left(x\right)=\frac{1}{(r-1)!}{\int }_a^b(x-t{)}^{r-1}{\phi }_n(t)dt, n=0,1,2,\dots .$

We study the conditions on a function $f\left(x\right)$, 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 ${\left\{{\psi }_{(}r,2n\left)\right(x)\right\}}_{n\ge 0}$. We also discuss the asymptotic properties of these functions, and this concludes our contribution.