Dunkl generalization of Phillips operators and approximation in weighted spaces

Advances in Difference Equations

article

Mursaleen, M.¹  Nasiruzzaman, Md.⁴  Kılıçman, A.⁵  Sapar, S. H.⁵ 

[1] Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan);Department of Mathematics, Aligarh Muslim University;Department of Computer Science and Information Engineering, Asia University;Department of Mathematics, Faculty of Science, University of Tabuk;Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia

关键词: Szász operator;    Dunkl analogue;    Generalization of exponential function;    Korovkin type theorem;    Modulus of continuity;    Order of convergence;   

DOI  :  10.1186/s13662-020-02820-9

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

The purpose of this article is to introduce a modification of Phillips operators on the interval$[ \frac{1}{2},\infty ) $ via a Dunkl generalization. We further define the Stancu type generalization of these operators as$\mathcal{S}_{n, \upsilon }^{\ast }(f;x)=\frac{n^{2}}{e_{\upsilon }(n\chi _{n}(x))}\sum_{\ell =0}^{\infty } \frac{(n\chi _{n}(x))^{\ell }}{\gamma _{\upsilon }(\ell )}\int _{0}^{\infty } \frac{e^{-nt}n^{\ell +2\upsilon \theta _{\ell }-1}t^{\ell +2\upsilon \theta _{\ell }}}{\gamma _{\upsilon }(\ell )}f ( \frac{nt+\alpha }{n+\beta } ) \,\mathrm{d}t$ ,$f\in C_{\zeta }(R^{+})$ , and calculate their moments and central moments. We discuss the convergence results via Korovkin type and weighted Korovkin type theorems. Furthermore, we calculate the rate of convergence by means of the modulus of continuity, Lipschitz type maximal functions, Peetre’s K-functional and the second order modulus of continuity.

【 授权许可】

CC BY