Some Results on Post-Widder Operators Preserving Test Function $x^r$
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Authors: V. GUPTA AND G. TACHEV
DOI: 10.46793/KgJMat2201.149G
Abstract:
In the present paper, we consider Post-Widder operators and its modified form which preserve the test function xr, r ∈ ℕ. We estimate direct results in terms of modulus of continuity for the modified operators. Also, some estimates for polynomially bounded functions and linear combinations are considered. Our estimates improve in some sense the previous results for the original Post-Widder operators.
Keywords:
Post-Widder operators, exponential functions, moments, polynomially bounded functions, linear combinations.
References:
[1] T. Acar, A. Aral and I. Rasa, The new forms of Voronovskaja’s theorem in weighted spaces, Positivity 20 (2016), 25–40.
[2] A. Aral, H. Gonska, M. Heilmann and G. Tachev, Quantitative Voronovskaya-type results for polynomially bounded functions, Results Math. 70(3–4) (2016), 313–324.
[3] D. Aydin, A. Aral and G. B-Tunca, A generalization of Post-Widder operators based on q-integers, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), DOI 10.2478/aicu-2014-0012.
[4] J. Bustamante, J. M. Quesada and L. M. de la Cruz, Direct estimate for positive linear operators in polynomial weighted spaces, J. Approx. Theory 162(8) (2010), 1495–1508.
[5] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, New York, 1987.
[6] B. R. Draganov and K. G. Ivanov, A characterization of weighted approximation by the Post-Widder and the Gamma operators, J. Approx. Theory 146 (2007), 3–27.
[7] B. R. Draganov and K. G. Ivanov, A characterization of weighted approximation by the Gamma and the Post-Widder operators, in: B. Bojanov, M. Drinov (Eds.), Proceedings of the International Conference on “Constructive Theory of Functions, Varna 2005”, Academic Publishing House, Sofia, 2006, 80–87.
[8] H. Gonska, Quantitative Aussagen zur Approximation durch positive lineare Operatoren, Ph. D. Thesis, Universität Duisburg, Duisburg, 1979.
[9] V. Gupta and G. Tachev, Approximation with Positive Linear Operators and Linear Combinations, Series: Developments in Mathematics 50, Springer, Cham, 2017.
[10] V. Gupta and G. Tachev, Approximation by Szász-Mirakyan Baskakov operators, J. Appl. Funct. Anal. 9(3-4) (2014), 308–319.
[11] N. Ispir, On modified Baskakov operatos on weighted spaces, Turk. J. Math. 25 (2001), 355–365.
[12] S. Li and R. T. Wang, The characterization of the derivatives for linear combinations of Post Widder operators in Lp, J. Approx. Theory 97 (1999), 240–255.
[13] C. P. May, Saturation and inverse theorems for combinations of a class of exponential-type operators, Canad. J. Math. 28 (1976), 1224–1250.
[14] R. Pǎltǎnea, Estimates of approximation in terms of a weighted modulus of continuity, Bull. Transilv. Univ. Brasov 4(53) (2011), 67–74.
[15] R. K. S. Rathore and O. P. Singh, On convergence of derivatives of Post-Widder operators, Indian J. Pure Appl. Math. 11 (1980), 547–561.
[16] L. Rempulska and M. Skorupka, On Approximation by Post-Widder and Stancu operators Preserving x2, Kyungpook Math. J. 49 (2009), 57–65.
[17] G. Tachev and V. Gupta, General form of Voronovskaja’s theorem in terms of weighted modulus of continuity, Results Math. 69(3-4) (2016), 419–430.
[18] D. V. Widder, The Laplace Transform, Princeton Mathematical Series, Princeton University Press, Princeton, New Jersey, 1941.
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