This article is cited in 11 papers

Some Problems of Approximation Theory in the Spaces $L_p$ on the Line with Power Weight

Iong Ping Li^a, Chun Mei Su^a, V. I. Ivanov<sup>b

a</sup> Beijing Normal University
^b Tula State University

Abstract: In the spaces $L_p$ on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the $K$-functional. We prove a direct and an inverse theorem of Jackson–Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the $K$-functional.

Keywords: Dunkl difference-differential operator, entire function, Dunkl transform, generalized shift operator, modulus of smoothness, the spaces $L_p$, Jackson–Stechkin theorem.

UDC: 517.5

Received: 23.03.2011
Revised: 04.06.2011

DOI: 10.4213/mzm9224

 English version: Mathematical Notes, 2011, 90:3, 344–364

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