Isaac Scientific Publishing

Advances in Analysis

Gegenbauer Transformations Nikolski-Besov Spaces Generalized by Gegenbauer Operator and Their Approximation Characteristics

Download PDF (686.8 KB) PP. 167 - 195 Pub. Date: May 3, 2017

DOI: 10.22606/aan.2017.23004

Author(s)

  • V.S. Guliyev*
    Ahi Evran University, Department of Mathematics, 40100, Kirsehir, Turkey
  • E.J. Ibrahimov

    Institute of Mathematics and Mechanics, AZ 1141 Baku, Azerbaijan
  • S.Ar. Jafarova

    Azerbaijan State Economic University 6, Istiglaliyyat str., Baku AZ1001, Azerbaijan

Abstract

Approximation of functions, generalized Gegenbauer shift, Gegenbauer transformation, Nikol’skii-Besov type spaces, embedding theorems.

Keywords

In this paper we consider some problems of the theory of approximation of functions on interval [0, ∞) in the metric of Lp, λ with weight sh χ using generalized Gegenbauer shifts. We prove analogues of direct Jackson theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Gegenbauer shifts. We establish the equivalence of the modulus of smoothness and K-functional, defined in terms of the space of the Sobolev type corresponding to the Gegenbauer differential operator. We define function spaces of Nikol’skii-Besov type and describe them in terms of best approximations. As a tool for approximation, we use some functions classes of spectrum. In these classes, we prove analogues of Bernstein’s inequality and others for the Gegenbauer differential operator. Our results are analogues of the results for generalized Bessel shifts obtained in the work [30].

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