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 Probl. Anal. Issues Anal., 2019, Volume 8(26), Issue 1, Pages 32–46 (Mi pa256)

Sobolev-orthonormal system of functions generated by the system of Laguerre functions

Dagestan Scientific Center of RAS, 45, M.Gadzhieva st., Makhachkala, 367025, Russia

Abstract: We consider the system of functions $\lambda_{r,n}^\alpha(x)$ ($r\in\mathbb{N}$, $n=0, 1, 2, \ldots$), orthonormal with respect to the Sobolev-type inner product $\langle f, g\rangle=\sum_{\nu=0}^{r-1}f^{(\nu)}(0)g^{(\nu)}(0)+\int_{0}^{\infty} f^{(r)}(x)g^{(r)}(x) dx$ and generated by the orthonormal Laguerre functions. The Fourier series in the system $\{\lambda_{r,n}^{\alpha}(x)\}_{k=0}^\infty$ is shown to uniformly converge to the function $f\in W_{L^p}^r$ for $\frac{4}{3}<p<4$, $\alpha\geq0$, $x\in[0, A]$, $0\leq A<\infty$. Recurrence relations are obtained for the system of functions $\lambda_{r,n}^\alpha(x)$. Moreover, we study the asymptotic properties of the functions $\lambda_{1,n}^\alpha(x)$ as $n\rightarrow\infty$ for $0\leq x\leq\omega$, where $\omega$ is a fixed positive real number.

Keywords: Laguerre polynomials, Laguerre functions, inner product of Sobolev type, Sobolev-orthonormal functions, recurrence relations, Fourier series, asymptotic formula.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-31-00477_mol_a This work was written with the support of the Russian Foundation for Basic Research (grant 18-31-00477_mol_a).

DOI: https://doi.org/10.15393/j3.art.2019.5150

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Bibliographic databases:

UDC: 517.521
MSC: 42C10, 65Q30
Revised: 04.02.2019
Accepted:03.02.2019
Language:

Citation: R. M. Gadzhimirzaev, “Sobolev-orthonormal system of functions generated by the system of Laguerre functions”, Probl. Anal. Issues Anal., 8(26):1 (2019), 32–46

Citation in format AMSBIB
\Bibitem{Gad19} \by R.~M.~Gadzhimirzaev \paper Sobolev-orthonormal system of functions generated by the system of Laguerre functions \jour Probl. Anal. Issues Anal. \yr 2019 \vol 8(26) \issue 1 \pages 32--46 \mathnet{http://mi.mathnet.ru/pa256} \crossref{https://doi.org/10.15393/j3.art.2019.5150} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000459770700003}