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Prakash, Chandra; Verma, Durvesh Kumar; Deo, Naokant

Approximation by a new sequence of operators involving Apostol-Genocchi polynomials. (English) Zbl 1491.41004

Math. Slovaca 71, No. 5, 1179-1188 (2021).
Summary: The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discuss the approximation results.

Cited in 4 Documents

MSC:

41A10 Approximation by polynomials
41A36 Approximation by positive operators
41A81 Weighted approximation

Keywords:

Apostal-Genocchi polynomials; asymptotic formula; rate of convergence
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Full Text: DOI

References:

[1] Abramowitz, M.—Stegun, I. A.: Handbook of Mathematical Functions, National Bureau of Standards, Washington, D. C., 1964. · Zbl 0171.38503
[2] Araci, S.: Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Appl. Math. Comput. 233 (2014), 599-607. · Zbl 1334.05013
[3] Claesson, A.—Kitaev, S.—Ragnarsson, K.—Tenner, B. E.: Boolean complexes for Ferrers graphs, Australas. J. Combin. 48 (2010), 159-173. · Zbl 1232.05009
[4] Chauhan, R.—Baxhaku, B.—Agrawal, P. N.: Szász Type Operators involving Charlier Polynomials of blending type, Complex Anal. Oper. Theory 13 (2019), 1197-1226. · Zbl 1426.41013
[5] De Vore, R. A.—Lorentz, G. G.: Constructive Approximation, Springer, Berlin 1993. · Zbl 0797.41016
[6] Deo, N.—Dhamija, M.—Miclǎuş, D.: Stancu-Kantorovich operators based on inverse Pólya-Eggenberger distribution, Appl. Math. Comput. 273(1) (2016) 281-289. · Zbl 1410.41031
[7] Deo, N.—Dhamija, M.: Generalized positive linear operators based on PED and IPED, Iranian J. Sci. Technol. Trans. Sci. 43 (2019), 507-513.
[8] Gadzjiv, A. D.: Theorems of the type of P.P. Korovkin type theorems, Math. Zametki 20(5) (1976), 781-786; Math. Notes 20(5-6) (1976), 996-998 (English translation).
[9] Gupta, V.—LÓpez-Moreno, A.-J.: Phillips operators preserving arbitrary exponential functions, e^at, e^bt, Filomat 32 (2018), 5071-5082. · Zbl 1499.41062
[10] Gupta, V.—Agarwal, R. P.: Convergence Estimates in Approximation Theory, Springer International Publishing, Switzerland, 2014. · Zbl 1295.41002
[11] Gupta, V.—Rassias, M. Th.: Moments of Linear Positive Operators and Approximation, Springer International Publishing, Switzerland, 2019. · Zbl 1448.41001
[12] Ispir, N.—Aral, A.—Dogru, O.: On Kantorovich process of a sequence of generalized positive linear operators, Nonlinear Funct. Anal. Optimiz. 29(5-6) 2008, 574-589. · Zbl 1152.41013
[13] Jolany, H.—Sharifi, H.—Alikelaye, R. E.: Some results for the Apostol-Genocchi polynomials of higher order, Bull. Malays. Math. Sci. Soc. 36(2) (2013), 465-479. · Zbl 1308.11023
[14] Jung, H. S.—Deo, N.—Dhamija, M.: Pointwise approximation by Bernstein type operators in mobile interval, Appl. Math. Comput. 214(1) (2014), 683-694. · Zbl 1335.41004
[15] Kajla, A.—Agrawal, P. N.: Szász-Kantorovich type operators based on Charlier polynomials, Kyungpook Math. J. 56(3) (2016), 877-897. · Zbl 1379.26006
[16] Kajla, A.—Agrawal, P. N.: Approximation properties of Szász type operators based on Charlier polynomials, Turkish J. Math. 39(6) (2015), 990-1003. · Zbl 1347.41028
[17] Korovkin, P. P.: On convergence of linear positive operators in the space of continuous functions (Russian), Doklady Akad. Nauk. SSSR (NS) 90 (1953), 961-964. · Zbl 0050.34005
[18] Luo, Q. M.: q-Extensions for the Apostol-Genocchi polynomials, Gen. Math. 17(2) (2009), 113-125. · Zbl 1189.11016
[19] Luo, Q. M.: Extensions of the Genocchi polynomials and their fourier expansions and integral representations, Osaka J. Math. 48 (2011), 291-309. · Zbl 1268.33002
[20] Luo, Q. M.—Srivastava, H. M.: Some generalizations of the-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011), 5702-5728. · Zbl 1218.11026
[21] Natanson, I. P.: Constructive Function Theory, Vol. 1, Fredric Unger, New York, 1964. · Zbl 0133.31101
[22] Ozden, H.—Simsek, Y.: Modification and unification of the Apostol-type numbers and polynomials and their applications, Appl. Math. Comput. 235 (2014), 338-351. · Zbl 1334.11015
[23] Remmel, J.: Ascent Sequences, 2 + 2-free Posets, Upper Triangular Matrices, and Genocchi numbers, Workshop on Combinatorics, Enumeration, and Invariant Theory, George Mason University, Virginia, 2010.
[24] Srivastava, H. M.—Özarslan, M. A.—Kaanoğlu, C.: Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Russ. J. Math. Phys. 20(1) (2013), 110-120. · Zbl 1292.11043
[25] Varma, S.—Taşdelen, F.: Szász type operators involving Charlier polynomials, Math. Comput. Modell. 56 (2012), 118-122. · Zbl 1255.41014
[26] Verma, D. K.—Gupta, V.: Approximation by a New Sequence of Operators Involving Charlier Polynomials with a Certain Parameter. In: Modern Mathematical Methods and High Performance Computing in Science and Technology, Springer, Singapore, 2016.
[27] Verma, D. K.—Gupta, V.—Agrawal, P. N.: Some approximation properties of Baskakov-Durrmeyer-Stancu operators, Appl. Math. Comput. 218(11) (2012), 6549-6556. · Zbl 1244.45005
[28] Weierstrass, K. G.: Über die analytische Darstellbarkeit sogenannter willkürlicher Funktionen einer reellen Veränderlichen, Sitzungsber. Akad. Berlin (1885), 633-639, 789-805. · JFM 17.0384.02
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