Approximation by a Composition of Apostol-Genocchi and P\v{a}lt\v{a}nea-Durrmeyer Operators
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Authors: N. S. MISHRA AND N. DEO
DOI: 10.46793/KgJMat2404.629M
Abstract:
The present paper deals with the Durrmeyer construction of operators based on a class of orthogonal polynomials called Apostol-Genocchi polynomials. For the proposed operators, we first establish a global approximation result followed by its convergence estimate in terms of usual, r-th and weighted modulus of continuity. We further study the asymptotic type results such as the Voronovskaya theorem and quantitative Voronovskaya theorem. Moreover, we estimate the rate of pointwise convergence of the proposed operators for functions of bounded variation defined on the interval (0,∞). Finally, the results are validated through graphical representations and an absolute error table.
Keywords:
Apostol-Genocchi polynomials, Pǎltǎnea basis, generating functions, special functions, functions of bounded variation.
References:
[1] T. Acar, A. Aral and I. Raşa, The new forms of Voronovskayas theorem in weighted spaces, Positivity 20(1) (2016), 25–40. https://doi.org/10.1007/s11117-015-0338-4
[2] A. M. Acu, T. Acar and V. A. Radu, Approximation by modified Unρ operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.). Serie A. Matemáticas 113(3) (2019), 2715–2729. https://doi.org/10.1007/s13398-019-00655-y
[3] A. M. Acu and V. A. Radu, Approximation by certain operators linking the α-Bernstein and the Genuine α-Bernstein-Durrmeyer operators, in: N.Deo et al. (Eds.), Mathematical Analysis I: Approximation Theory, Springer, Proceedings in Mathematics & Statistics 306, 2020, 77–88.
[4] I. N. Cangul, H. Ozden and Y. Simsek, A new approach to q-Genocchi numbers and their interpolation functions, Nonlinear Analysis: Theory, Methods & Applications 71(12) (2009), e793–e799. https://doi.org/10.1016/j.na.2008.11.040
[5] N. Deo and S. Kumar, Durrmeyer variant of Apostol-Genocchi-Baskakov operators, Quaest. Math. 44(4) (2020), 1–18. https://doi.org/10.2989/16073606.2020.1834000
[6] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993.
[7] M. Dhamija, Durrmeyer modification of Lupas type Baskakov operators based on IPED, in: N. Deo et al. (Eds.), Mathematical Analysis I: Approximation Theory, Springer, Proceedings in Mathematics & Statistics 306, 2020, 111–120.
[8] M. Dhamija and N. Deo, Jain-Durrmeyer operators associated with the inverse Pólya–Eggenberger distribution, Appl. Math. Comput. 286 (2016), 15–22. https://doi.org/10.1016/j.amc.2016.03.015
[9] A. D. Gadjiev, Theorems of the type of PP Korovkins theorems, Mat. Zametki. 20(5) (1976), 781–786.
[10] T. Garg, A. M. Acu and P. N. Agrawal, Further results concerning some general Durrmeyer type operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(3) (2019), 2373–2390. https://doi.org/10.1007/s13398-019-00628-1
[11] V. Gupta, A note on the general family of operators preserving linear functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(4) (2019), 3717–3725. https://doi.org/10.1007/s13398-019-00727-z
[12] M. Heilmann and I. Raşa, A nice representation for a link between Baskakov and Szász-Mirakjan-Durrmeyer operators and their Kantorovich variants, Results Math. 74(1) (2019), 1–12. https://doi.org/10.1007/s00025-018-0932-4
[13] A. Kajla, N. Ispir, P. N. Agrawal and M. Goyal, q-Bernstein-Schurer-Durrmeyer type operators for functions of one and two variables, Appl. Math. Comput. 275 (2016), 372–385. https://doi.org/10.1016/j.amc.2015.11.048
[14] P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publication Co., Delhi 1960.
[15] Q. M. Luo, q-Extensions for the Apostol-Genocchi polynomials, General Mathematics 17(2) (2009), 113–125.
[16] Q. M. Luo, Extensions of the Genocchi polynomials and their Fourier expansions and integral representations, Osaka J. Math. 48 (2011), 291–309.
[17] Q. M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217(12) (2011), 5702–5728. https://doi.org/10.1016/j.amc.2010.12.048
[18] M. A. Özarslan, Unified Apostol-Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 62(6) (2011), 2452–2462. https://doi.org/10.1016/j.camwa.2011.07.031
[19] R. Pǎltǎnea, Modified Szász-Mirakjan operators of integral form, Carpathian J. Math. 24(3) (2008), 378–385.
[20] C. Prakash, D. K. Verma and N. Deo, Approximation by a new sequence of operators involving Apostol-Genocchi polynomials, Math. Slovaca (to appear).
[21] H. M. Srivastava, B. Kurt and V. Kurt, Identities and relations involving the modified degenerate hermite-based Apostol-Bernoulli and Apostol-Euler polynomials, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.). Serie A. Matemáticas 113(2) (2019), 1299–1313. https://doi.org/10.1007/s13398-018-0549-1
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