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

Abstract

In this paper we consider some problems of the theory of approximation of functions on interval [0, ∞) in the metric of L p,λ with weight sh 2λ x 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].Keywords: Approximation of functions, generalized Gegenbauer shift, Gegenbauer transformation, Nikol'skii-Besov type spaces, embedding theorems.In the classical theory of approximation of functions on R = (−∞, ∞) the shift operator f (x) → f (x + y), x, y ∈ R. plays a central role. This shift operator is used in the construction of the moduli of continuity and smoothness, which are the basic elements of the direct and inverse theorems of approximation theory. Various generalizations of shift operators enable to state natural analogues of problems in approximation theory. Groups and semigroups of operators on Banach spaces are generalizations of the shift operator. Different problems of approximation theory on Banach spaces with groups and semigroups of operators were considered in [1,3,5,7,44].Generalized shift operator naturally follows from "addition theorem" for eigen functions of the differential operators (for example, Legendre, Gegenbauer, Jacobi, Laguerre, Hermite and other). These operators may not form a group or semigroup, but the generalized moduli of smoothness defined in terms of them can be better adapted to the study of relations between the smoothness properties of functions and the best approximations of these functions in weighted function spaces. Some results on the best approximation of functions using generalized shift operators can be found in [1, 4, 22-26, 29-39, 45]. Note that most of the papers on this topic deal with the approximation of functions by polynomials on a finite segment. For the half-line, most popular examples are the generalized Bessel and Dunkl shifts (see for example [4,23,[30][31][32]). Fourier-Bessel and Fourier-Dunkl harmonic analysis, which deals with Bessel and Dunkl integral transformations and their applications, are closely connected with the generalized Bessel and Dunkl shift. Moreover, generalized Bessel shift is widely used in the potential theory and theory of maximal functions (see, example [5,[11][12][13]). The file of constructions of the theory of generalized shift operators generalize in the theory of transformation of operators (see for example [7]). We reduce only ...