Approximation by generalized Bernstein–Stancu operators

In this paper, we investigate approximation properties of the Stancu type generalization of the -Bernsteinoperator. We obtain a recurrence relation for moments and the rate of convergence by means of moduli of continuity.Also, we present Voronovskaya and Grüss–Voronovskaya type asymptotic results for these operators. Finally, the studycontains numerical considerations regarding the constructed operators based on Maple algorithms.

PDF

___

[1] Acar T, Acu AM, Manav N. Approximation of functions by genuine Bernstein-Durrmeyer type operators. Journal of Mathematical Inequalities 2018; 12 (4): 975-987.
[2] Acar T, Aral A, Raşa I. Positive linear operators preserving and 2 . Constructive Mathematical Analysis 2019; 2 (3): 98-102.
[3] Acar T, Kajla A. Degree of approximation for bivariate generalized Bernstein type operators. Results in Mathematics 2018; 73: 79. doi: 10.1007/s00025-018-0838-1
[4] Acu AM, Acar T, Radu VA. Approximation by modified U n operators. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A Matemáticas 2019; 113: 2715-2729.
[5] Acu AM, Gonska HH, Raşa I. Grüss-type and Ostrowski-type inequalities in approximation theory. Ukrainian Mathematical Journal 2011; 63 (6): 843-864.
[6] Agratini O. Linear combinations of D.D. Stancu polynomials. Revista de Analiză Numerică și Teoria Aproximației 1998; 27: 15-22.
[7] Altomare F, Campiti M. Korovkin-type Approximation Theory and its Applications. New York, NY, USA: Walter de Gruyter, 1994.
[8] Bodur M, Yilmaz OG, Aral A. Approximation by Baskakov-Szász-Stancu operators preserving exponential functions. Constructive Mathematical Analysis 2018; 1 (1): 1-8.
[9] Campiti M. A generalization of Stancu-Mulbach operators. Constructive Approximation 1991; 7 (1): 1-18.
[10] Çetin N. Approximation by complex modified Szasz-Mirakjan-Stancu operators in compact disks. Filomat 2015; 29 (5): 1007-1019.
[11] Çetin N. On complex modified genuine Szasz-Durrmeyer-Stancu operators. Communications of the Faculty of Sciences of University of Ankara Series A1-Mathematics and Statistics 2019; 68 (1): 283-298.
[12] Chen X, Tan J, Liu Z, Xie J. Approximation of functions by a new family of generalized Bernstein operators. Journal of Mathematical Analysis and Applications 2017; 450: 244-261.
[13] Cleciu VA. Bernstein-Stancu operators. Studia Universitatis Babes-Bolyai Mathematica 2007; 52 (4): 53-65.
[14] Cleciu VA. Approximation properties of a class of Bernstein-Stancu type operators. In: Agratini O (editor). Proceedings of the International Conference NAAT; Cluj-Napoca, Romania; 2006. pp. 171-178.
[15] Della Vechia B. On Stancu operator and its generalizations. Naples, Italy: Inst. Applicazioni della Matematica (Rapp. Tecnico), 1988.
[16] Gal S, Gupta V. Approximation by complex Szasz-Mirakyan-Stancu-Durrmeyer operators in compact disks under exponential growth. Filomat 2015; 29 (5): 1127-1136.
[17] Gavrea I. On Bernstein-Stancu type operators. Studia Universitatis Babes-Bolyai 2007; 52 (4): 81-88.
[18] Gonska HH, Piţul P, Raşa I. Over-iterates of Bernstein-Stancu operators. Calcolo 2007; 44 (2): 117-125.
[19] Gonska HH, Tachev G. Grüss-type inequalities for positive linear operators with second order moduli. Matematicki Vesnik 2011; 63 (4): 247-252.
[20] Ibrahim RW. Geometric properties of the complex Baskakov-Stancu operators in the unit disk. Boletim Sociedade Paranaense de Matematica 2015; 33 (1): 23-32.
[21] Kageyama Y. A new class of modified Bernstein operators. Journal of Approximation Theory 1999; 101: 121-147.
[22] Kajla A. On the Bézier variant of the Srivastava-Gupta operators. Constructive Mathematical Analysis 2018; 1 (2): 99-107.
[23] Kajla A, Acar T. A new modification of Durrmeyer type mixed hybrid operators. Carpathian Journal of Mathematics 2018; 34 (1): 47-56.
[24] Kajla A, Acar T. Blending type approximation by generalized Bernstein-Durrmeyer type operators. Miskolc Mathematical Notes 2018; 19 (1): 319-336.
[25] Kumar AS, Acar T. Approximation by generalized Baskakov-Durrmeyer-Stancu type operators. Rendiconti del Circolo Matematico di Palermo Series 2016; 65 (3): 411-424.
[26] Kwun YC, Acu AM, Rafiq A, Radu VA, Ali F et al. Bernstein-Stancu type operators which preserve polynomials. Journal of Computational Analysis and Applications 2017; 23 (1): 758-770.
[27] Lupaş A. Contributions to the theory of approximation by linear operators. PhD, University of Babeş-Bolyai, Cluj-Napoca, Romania, 1976.
[28] Mastroianni G, Occorsio MR. Una generalizzazione dell’operatore di Stancu. Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche 1978; 45 (4): 495-511 (in Italian).
[29] Muraru CV, Acu AM, Radu VA. On the monotonicity of q-Schurer-Stancu type polynomials. Miskolc Mathematical Notes 2018; 19 (1): 19-28.
[30] Păltănea R. Approximation Theory Using Positive Linear Operators. Boston, MA, USA: Birkhauser, 2004
31] Radu VA. Quantitative estimates for some modified Bernstein-Stancu operators. Miskolc Mathematical Notes 2018; 19 (1): 517-525.
[32] Ren MY, Zeng XM. Approximation by complex Stancu type Durrmeyer operators in compact disks. Journal of Computational Analysis and Applications 2015; 18 (3): 440-453.
[33] Stancu DD. Approximation of functions by a new class of linear polynomial operators. Revue Roumaine de Mathématique Pures et Appliquées 1968; 13: 1173-1194.
[34] Stancu DD. Asupra unei generalizări a polinoamelor lui Bernstein. Studia Universitatis Babeş-Bolyai 1969; 14 (2): 31-45 (in Romanian).

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv