On Complex Modified Genuine Szász-Durrmeyer-Stancu Operators

Yıl 2019, Cilt: 68 Sayı: 1, 283 - 298, 01.02.2019

Nursel Çetin

https://doi.org/10.31801/cfsuasmas.415926

Öz

In this paper, we introduce complex modified genuine Szász-Durrmeyer-Stancu operators to improve the results obtained in [4] and present overconvergence properties of these operators. We obtain some estimates on the rate of convergence, a Voronovskaja-type result and the exact order of approximation for these operators attached to analytic functions of exponential growth on compact disks.

Anahtar Kelimeler

Complex genuine Szász-Durrmeyer-Stancu operators, overconvergence, simultaneous approximation, Voronovskaja-type result, exact order of approximation

Kaynakça

• Agrawal P.N. and Gupta V., On convergence of derivatives of Phillips operators, Demonstratio Math. 27 (2), 501-510 (1994).
• Bernstein S. N., Sur la convergence de certaines suites des polynomes, J. Math. Pures Appl. 15(9), 345-358 (1935).
• Çetin N., Approximation by complex modified Szász-Mirakjan-Stancu operators in compact disks, Filomat, 29 (5), 1007-1019 (2015).
• Çetin N. and İspir N., Approximation properties of complex modified genuine Szász-Durrmeyer operators, Comput. Methods Funct. Theory (2014) 14:623-638.
• Deeba E. Y., On the convergence of generalized Szasz operator on complex plane, Tamkang J. Math. 13 (1982), no. 1, 79--86.
• Finta Z., Gupta V., Direct and inverse estimates for Phillips type operators, J. Math. Anal. Appl. 303 (2) (2005), 627-642.
• Gal S. G., Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, 8. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009. xii+337 pp. ISBN: 978-981-4282-42-0.
• Gal S. G., Approximation by complex genuine Durrmeyer type polynomials in compact disks, Appl. Math. Comput. 217 (2010) 1913-1920.
• Gal S. G., Gupta V., Approximation by Complex Durrmeyer Type Operators in Compact Disks, Mathematics Without Boundaries, in:P. Pardalos, T.M. Rassias (Eds.), Surveys in Interdisciplinary Research, Springer, 2014.
• Gergen J. J., Dressel F. G. and Purcell W.H., Convergence of extended Bernstein polynomials in the complex domain, Pacific J. Math. 13 (4) (1963) 1171-1180.
• Gupta V., Tachev G. , Approximation by linear combinations of complex Phillips operators in compact disks, Results. Math., DOI 10.1007/s00025-014-0377-3.
• Gupta V. and Verma D. K., Approximation by complex Favard-Szász-Mirakjan-Stancu operators in compact disks, Mathematical Sciences 6:25 (2012).
• Gupta V., Complex Baskakov-Szász operators in compact semi-disks, Lobachevskii J. Math. 35(2), 65-73 (2014).
• Heilmann M., Tachev G., Commutativity, direct and strong converse results for Phillips operators, East J. Approx. 17(3) (2011) 299-317.
• Jakimovski A. and Leviatan D., Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 34 (1969), 97-103.
• Kantorovich L. V., Sur la convergence de la suite de polynomes de S. Bernstein en dehors de l'interval fundamental. Bull. Acad. Sci. URSS 1103-1115 (1931).
• Lorentz G. G., Bernstein Polynomials, 2nd ed., Chelsea Publ., New York (1986).
• May C.P., On Phillips operator, J. Approx. Theory 20, 315-332 (1977).
• Mazhar S.M., Totik V., Approximation by modified Szász operators, Acta Sci. Math. 49 (1985), 257-269.
• Phillips R. S., An inversion formula for semi-groups of linear operators, Ann. Math. 59 (1954) 352--356.
• Tonne, P.C., On the convergence of Bernstein polynomials for some unbounded analytic functions, Proc. Am. Math. Soc., 22,1-6.
• Wood B., Generalized Szász operators for the approximation in the complex domain, SIAM J. Appl. Math., 17(4), (1969), 790--801.
• Wright E. M., The Bernstein approximation polynomials in the complex plane, J. Lond. Math. Soc. 5, 265-269 (1930).