Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Gadzhimirzaev R. M. The Fourier Series of the Meixner Polynomials Orthogonal with Respect to the Sobolev-type Inner Product. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 4, pp. 388-395. DOI: 10.18500/1816-9791-2016-16-4-388-395, EDN: XHPYGJ

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
14.11.2016
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Russian
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UDC: 
517.52
EDN: 
XHPYGJ

The Fourier Series of the Meixner Polynomials Orthogonal with Respect to the Sobolev-type Inner Product

Autors: 
Gadzhimirzaev Ramis Makhmudovich, Daghestan Scientific Centre of Russian Academy of Sciences
Abstract: 

In this paper we consider the system of discrete functions {ϕr,k(x)} ∞ k=0 , which is orthonormal with respect to the Sobolev-type inner product hf, gi = Xr−1 ν=0 ∆ ν f(−r)∆ν g(−r) + X t∈Ωr ∆ r f(t)∆r g(t)µ(t), where µ(t) = q t (1−q), 0 < q < 1. It is shown that the shifted classical Meixner polynomials © M−r k (x + r) ª∞ k=r together with functions n (x+r) [k] k! or−1 k=0 form a complete orthogonal system in the space l2,µ(Ωr) with respect to the Sobolev-type inner product. It is shown that the Fourier series on Meixner polynomials © akM−r k (x + r) ª∞ k=r (ak— normalizing factors), orthonormal in terms of Sobolev, is a special case of mixed series on Meixner polynomials. Some new special series on Meixner orthogonal polynomials Mα k (x) with α > −1 are considered. In the case when α = r these special series coincide with mixed series on Meixner polynomials M0 k (x) and Fourier series on the system © akM−r k (x + r) ª∞ k=r orthonormal with respect to the Sobolev-type inner product. 

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Received: 
15.07.2016
Accepted: 
23.10.2016
Published: 
30.11.2016