Generalized Bernstein-Chlodowsky-Kantorovich type operators involving Gould-Hopper polynomials

Volume 14, Issue 5, pp 324--338 http://dx.doi.org/10.22436/jnsa.014.05.03

Publication Date: February 14, 2021 Submission Date: November 16, 2020 Revision Date: December 30, 2020 Accteptance Date: January 15, 2021

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Authors

P. N. Agrawal - Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India. Sompal Singh - Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India.

Abstract

In the present article, we establish a link between the theory of positive linear operators and the orthogonal polynomials by defining Bernstein-Chlodowsky-Kantorovich operators based on Gould-Hopper polynomials (orthogonal polynomials) and investigate the degree of convergence of these operators for unbounded continuous functions having a polynomial growth. In this connection, the moments of the operators are derived first, and then the approximation degree of the considered operators is established by means of the complete and the partial moduli of continuity. Next, we focus on the rate of convergence of these operators for functions in a weighted space. The associated Generalized Boolean Sum (GBS) operator of the operators under study is defined, and the degree of approximation is studied with the aid of the mixed modulus of smoothness and the Lipschitz class of Bögel continuous functions.

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Keywords

• Gould-Hopper polynomials
• modulus of continuity
• Peetre's K-functional
• Bögel continuous functions
• mixed modulus of smoothness

MSC

•  41A10
•  41A25
•  41A36
•  41A63

References

• [1] A. M. Acu, T. Acar, C.-V. Muraru, V. A. Radu, Some approximation properties by a class of bivariate operators, Math. Methods Appl. Sci., 42 (2019), 5551--5565
  • View Article


• [2] A. M. Acu, M. Dancs, V. A. Radu, Representations for the inverses of certain operators,, Commun. Pure Appl. Anal., 19 (2020), 4097--4109
  • View Article


• [3] P. N. Agrawal, B. Baxhaku and R. Chauhan, The approximation of bivariate Chlodowsky-Sz´asz-Kantorovich-Charliertype operators, J. Inequal. Appl., 2017 (2017), 23 pages
  • View Article


• [4] P. N. Agrawal, N. Ispir, Degree of approximation for bivariate Chlodowsky-Szasz-Charlier type operators, Results Math., 69 (2016), 369--385
  • View Article


• [5] C. Badea, I. Badea, C. Cottin, H. H. Gonska, Notes on the degree of approximation of B-continuous and B-differentiable functions,, J. Approx. Theory Appl., 4 (1988), 95--108
  • View Article


• [6] C. Badea, I. Badea, H. H. Gonska, A test function theorem and approximation by pseudopolynomials, Bull. Austral. Math. Soc., 34 (1986), 53--64
  • View Article


• [7] C. Badea, C. Cottin, Korovkin-type theorems for generalized Boolean sum operators, Approximation theory (Kecskemet,1990), Colloq. Math. Soc. Janos Bolyai, 58 (1991), 51--67
  • Google Scholar


• [8] B. Baxhaku, A. Berisha, The approximation Szasz-Chlodowsky type operators involving Gould-Hopper type polynomials, Abstr. Appl. Anal., 2017 (2017), 8 pages
  • View Article


• [9] K. Bogel, Mehrdimensionale Differentiation von Funktionen mehrerer reeller Veranderlichen, J. Reine Angew. Math., 170 (1934), 197--217
  • View Article


• [10] K. Bogel, Uber mehrdimensionale Differentiation, Integration und beschrankte Variation, J. Reine Angew. Math., 173 (1935), 5--30
  • View Article


• [11] P. L. Butzer, H. Berens, Semi-groups of Operators and Approximation, Springer-Verlag, New York (1967)
  • View Article


• [12] X. Chen, J. Tan, Z. Liu, J. Xie, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl., 450 (2017), 244--261
  • View Article


• [13] E. Dobrescu, I. Matei, The approximation by Bernˇste˘ın type polynomials of bidimensionally continuous functions, An. Univ. Timisoara Ser. Sti. Mat.-Fiz., 4 (1966), 85--90
  • Google Scholar


• [14] A. D. Gadziev, Positive linear operators in weighted spaces of functions of several variables (Russian), Izv. Akad. Nauk Azerbaıdzhan. SSR Ser. Fiz.-Tekhn. Mat. Nauk, 1 (1980), 32--37
  • Google Scholar


• [15] A. D. Gadziev, H. Hacisalihoglu, Convergence of the sequences of linear positive operators, Ankara University Press, Ankara, , Turkey (1995)
  • Google Scholar


• [16] V. Gupta, T. M. Rassias, P. N. Agrawal, A. M. Acu, Recent Advances in Constructive Approximation Theory,, Springer, Cham (2018)
  • View Article


• [17] N. Ispir, C. Atakut, Approximation by modified Szasz-Mirakjan operators on weighted spaces, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 571--578
  • View Article


• [18] A. Kajla, T. Acar, Modified α-Bernstein operators with better approximation properties, Ann. Funct. Anal., 10 (2019), 570--582
  • View Article


• [19] M. Sidharth, A. M. Acu, P. N. Agrawal, Chlodowsky-Szasz-Appell-type operators for functions of two variables, Ann. Funct. Anal., 8 (2017), 446--459
  • View Article