Abstract
This paper is intended to establish the equivalence between K-functionals and modulus of smoothness tied to a Dunkl type operator on the real line.
Highlights
Consider the first-order singular differential-difference operator on the real line Λf ( x=) f ′( x) + α + (x)− f x (−x) − 2n f, x where α > −1 2 and n = 0,1
How to cite this paper: Al Subaie, R.F. and Mourou, M.A. (2015) Equivalence of K-Functionals and Modulus of Smoothness Generated by a Generalized Dunkl Operator on the Real Line
The authors have developed in [7] [8] a new harmonic analysis on the real line related to the differential-difference operator Λ in which several classical analytic structures such as the Fourier transform, the translation operators, the convolution product, ..., were generalized
Summary
Consider the first-order singular differential-difference operator on the real line. , x where α > −1 2 and n = 0,1,. (2015) Equivalence of K-Functionals and Modulus of Smoothness Generated by a Generalized Dunkl Operator on the Real Line. The authors have developed in [7] [8] a new harmonic analysis on the real line related to the differential-difference operator Λ in which several classical analytic structures such as the Fourier transform, the translation operators, the convolution product, ..., were generalized. As the notion of translation operators was extended to various contexts (see [9] [10] and the references therein), many generalized modulus of smoothness have been discovered Such generalized modulus of smoothness are often more convenient than the usual ones for the study of the connection between the smoothness properties of a function and the best approximations of this function in weight functional spaces (see [11]-[13] and references therein).
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