A Kantorovich Type Generalization of the Szàsz Operators via Two Variable Hermite Polynomials

Yıl 2017, Cilt: 30 Sayı: 4, 432 - 440, 11.12.2017

Serdal Yazıcı , Bayram Çekim

Öz

The purpose of this paper is to give the Kantorovich generalization of the operators via two
variable Hermite polynomials which are introduced by Krech [1] and to research approximating
features with help of the classical modulus of continuity, the class of Lipschitz functions,
Voronovskaya type asymptotic formula, second modulus of continuity and Peetre's
K -
functional for these operators.

Anahtar Kelimeler

Hermite polynomial, Kantorovich type generalization, Modulus of continuity, Voronovskaya type asymptotic formula

Kaynakça

• [1] Krech, G., “A note on some positive linear operators associated with the Hermite polynomials’’, Carpathian J. Math., 32 (1): 71-77, (2016).
• [2] Szász, O., “Generalization of S. Bernstein’s polynomials to the infinite interval’’, J. Res. Nat. Bur. Stand., 45: 239-245, (1950).
• [3] Sucu, S., İçöz, G.,Varma, S., “On some extensions of Szász opertors including Boas-Buck type polynomials’’, Abstr. Appl. Anal.,Vol.2012, Article ID 680340: 15 pages, (2012).
• [4] Varma, S., Sucu, S., İçöz, G., “Generalization of Szász operators involving Brenke type polynomials’’, Comput. Math. Appl., 64(2): 121-127, (2012).
• [5] Varma, S., Taşdelen, F., “Szász type operators involving Charlier polynomials’’, Mathematical and Computer Modeling, 56: 118-122, (2012).
• [6] Atakut, Ç., Büyükyazici, İ., “Stancu type generalization of the Favard Szász operators’’, Appl. Math. Lett., 23(12): 1479-1482, (2010).
• [7] Ciupa, A., “A class of integral Favard-Szász type operators’’, Stud. Univ. Babes-Bolyai Math., 40(1): 39-47, (1995).
• [8] Gadzhiev, A. D.,“ The convergence problem for sequence of positive linear operators on unbounded sets and theorem analogues to that of P.P.Korovkin’’, Sov. Math. Dokl., 15(5): 1436-1453, (1974).
• [9] Stancu, D. D. “Approximation of function by a new class of polynomial operators’’, Rev. Rourn. Math. Pures et Appl., 13(8): 1173-1194, (1968).
• [10] Gupta, V., Vasishtha, V., Gupta, M. K., “Rate of convergence of the Szász-Kantorovich-Bezier operators for bounded variation function’’, Publ. Ins. Math.(Beograd)(N.S.), 72: 137-143, (2006).
• [11] Atakut, Ç., İspir, N., “Approximation by modified Szász-Mirakjan operators on weighted spaces’’, Proc. Indian Acad. Sci. Math., 112: 571-578, (2012).
• [12] Taşdelen, F., Aktaş, R., Altın, A., “A Kantorovich type of Szász operators including Brenke type polynomials’’, Abstract and Applied Analysis, Vol.2012: 13 pages, (2012).
• [13] Appell, P., Kampe de Feriet, J., Hypergeometriques et Hyperspheriques: Polynomes d’Hermite, Gauthier-Villars, Paris, 1926.
• [14] DeVore, R. A. and Lorentz, G.G., Constructive Approximation, Springer-Verlag, Berlin, 1993.
• [15] Ditzian, Z. and Totik, V., Moduli of smoothness, Springer-Verlag, New York, 1987.
• [16] Korovkin, P. P., “On convergence of linear positive operators in the space of continuous functions” (Russian), Doklady Akad. Nauk. SSSR (NS) 90: 961–964, (1953).
• [17] Özarslan, M. A. and Duman, O., “Approximation properties of Poisson integrals for orthogonal expansions”, Taiwanese J. Math. 12: 1147-1163, (2008).
• [18] Toczek, G. and Wachnicki, E., “On the rate of convergence and the Voronovskaya theorem for the Poisson integrals for Hermite and Laguerre expansions”, J. Approx. Theory, 116: 113-125, (2002).