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This article is cited in 5 scientific papers (total in 5 papers)
Sobolev-orthonormal system of functions generated by the system of Laguerre functions
R. M. Gadzhimirzaev 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.
Received: 02.11.2018 Revised: 04.02.2019 Accepted: 03.02.2019
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
Linking options:
https://www.mathnet.ru/eng/pa256 https://www.mathnet.ru/eng/pa/v26/i1/p32
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Abstract page: | 270 | Full-text PDF : | 55 | References: | 17 |
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