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
-
1020
Downloads
-
2206
Views
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.
Share and Cite
ISRP Style
P. N. Agrawal, Sompal Singh, Generalized Bernstein-Chlodowsky-Kantorovich type operators involving Gould-Hopper polynomials, Journal of Nonlinear Sciences and Applications, 14 (2021), no. 5, 324--338
AMA Style
Agrawal P. N., Singh Sompal, Generalized Bernstein-Chlodowsky-Kantorovich type operators involving Gould-Hopper polynomials. J. Nonlinear Sci. Appl. (2021); 14(5):324--338
Chicago/Turabian Style
Agrawal, P. N., Singh, Sompal. "Generalized Bernstein-Chlodowsky-Kantorovich type operators involving Gould-Hopper polynomials." Journal of Nonlinear Sciences and Applications, 14, no. 5 (2021): 324--338
Keywords
- Gould-Hopper polynomials
- modulus of continuity
- Peetre's K-functional
- Bögel continuous functions
- mixed modulus of smoothness
MSC
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
-
[2]
A. M. Acu, M. Dancs, V. A. Radu, Representations for the inverses of certain operators,, Commun. Pure Appl. Anal., 19 (2020), 4097--4109
-
[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
-
[4]
P. N. Agrawal, N. Ispir, Degree of approximation for bivariate Chlodowsky-Szasz-Charlier type operators, Results Math., 69 (2016), 369--385
-
[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
-
[6]
C. Badea, I. Badea, H. H. Gonska, A test function theorem and approximation by pseudopolynomials, Bull. Austral. Math. Soc., 34 (1986), 53--64
-
[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
-
[8]
B. Baxhaku, A. Berisha, The approximation Szasz-Chlodowsky type operators involving Gould-Hopper type polynomials, Abstr. Appl. Anal., 2017 (2017), 8 pages
-
[9]
K. Bogel, Mehrdimensionale Differentiation von Funktionen mehrerer reeller Veranderlichen, J. Reine Angew. Math., 170 (1934), 197--217
-
[10]
K. Bogel, Uber mehrdimensionale Differentiation, Integration und beschrankte Variation, J. Reine Angew. Math., 173 (1935), 5--30
-
[11]
P. L. Butzer, H. Berens, Semi-groups of Operators and Approximation, Springer-Verlag, New York (1967)
-
[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
-
[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
-
[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
-
[15]
A. D. Gadziev, H. Hacisalihoglu, Convergence of the sequences of linear positive operators, Ankara University Press, Ankara, , Turkey (1995)
-
[16]
V. Gupta, T. M. Rassias, P. N. Agrawal, A. M. Acu, Recent Advances in Constructive Approximation Theory,, Springer, Cham (2018)
-
[17]
N. Ispir, C. Atakut, Approximation by modified Szasz-Mirakjan operators on weighted spaces, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 571--578
-
[18]
A. Kajla, T. Acar, Modified α-Bernstein operators with better approximation properties, Ann. Funct. Anal., 10 (2019), 570--582
-
[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