On a new kind of λ-Bernstein-Kantorovich operators for univariate and bivariate functions

Cai Q., Kangal E., DİNLEMEZ KANTAR Ü., Zhou G., Aslan R.

Filomat, vol.39, no.5, pp.1437-1456, 2025 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 39 Issue: 5
  • Publication Date: 2025
  • Doi Number: 10.2298/fil2505437c
  • Journal Name: Filomat
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1437-1456
  • Keywords: basis function, Bernstein operators, Bernstein-Kantorovich operators, modulus of continuity, Peetre’s K-functional, rate of convergence, tensor product, Voronovskaja asymptotic formula
  • Van Yüzüncü Yıl University Affiliated: Yes

Abstract

This paper presents a novel class of λ-Bernstein operators, wherein the parameter λ ∈ [−1, 1]. An approximation theorem of the Korovkin type is explored, a local approximation theorem is established and an asymptotic formula of the Voronovskaja type is derived. In addition, the bivariate tensor product operators are built, some approximation properties are discussed, including an asymptotic theorem of the Voronovskaja type and the order of convergence in relation to Peetre’s K-functional. Finally, for certain continuous functions, numerical examples and plots to demonstrate our newly defined operators’ convergence behavior are provided and there are also provided in comparison with the classical Kantorovich operators in terms of the approximation error.