Relatively Uniform Weighted Summability Based on Fractional-Order Difference Operator


Kadak U., Srivastava H. M., Mursaleen M.

BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, vol.42, no.5, pp.2453-2480, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 42 Issue: 5
  • Publication Date: 2019
  • Doi Number: 10.1007/s40840-018-0612-2
  • Journal Name: BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2453-2480
  • Keywords: Weighted statistical convergence and weighted statistical summability, Fractional-order difference operators of functions, Relatively uniform convergence, The rates of convergence, Korovkin- and Voronovskaja-type approximation theorems, STATISTICAL CONVERGENCE, APPROXIMATION THEOREMS, SEQUENCE-SPACES, KOROVKIN, (P
  • Gazi University Affiliated: Yes

Abstract

In the present paper, we introduce the notion of relatively uniform weighted summability and its statistical version based upon fractional-order difference operators of functions. The concept of relatively uniform weighted alpha beta-statistical convergence is also introduced and some inclusion relations concerning the newly proposed methods are derived. As an application, we prove a general Korovkin-type approximation theorem for functions of two variables and also construct an illustrative example by the help of generating function type non-tensor Meyer-Konig and Zeller operators. Moreover, it is shown that the proposed methods are non-trivial generalizations of relatively uniform convergence which includes a scale function. We estimate the rate of convergence of approximating positive linear operators by means of the modulus of continuity and give a Voronovskaja-type approximation theorem. Finally, we present some computational results and geometrical interpretations to illustrate some of our approximation results.