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Approximation of functions and Mihesan operators

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  • For real numbers a,q0 and a weight ϱ(x)=1/(1+x)q, the author provides necessary and sufficient conditions for a function fC[0,) in order to supx0ϱ(x)(Ban(f,x)f(x))∣→0 as n, where Ban(f) is the Mihesan operator. In particular, it is proved that Mihesan operators behave similar to the classical Baskakov operators.

    Mathematics Subject Classification: Primary: 41A36; Secondary: 41A35.

    Citation:

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  • [1] P. N. AgrawalTh. A. K. Sinha and and A. Sharma, Convergence of derivative of Szász type operators involving Charlier polynomials, Math. Found. Computing, 5 (2022), 1-15.  doi: 10.3934/mfc.2021016.
    [2] M. Becker, Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27 (1978), 127-142.  doi: 10.1512/iumj.1978.27.27011.
    [3] J. BustamanteA. Carrillo-Zentella and J. M. Quesada, Direct and strong converse theorems for a sequence of positive linear operators, Acta Math. Hungar., 136 (2012), 90-106.  doi: 10.1007/s10474-012-0196-5.
    [4] J. Bustamante, J. J. Merino-García and J. M. Quesada, Baskakov operators and Jacobi weights: Pointwise estimates, Journal of Inequalities and Applications, (2021), Paper No. 119, 17 pp. doi: 10.1186/s13660-021-02653-4.
    [5] I. Gadjev, Weighted approximation by Baskakov operators, Math. Ineq. Appli., 18 (2015), 1443-1461.  doi: 10.7153/mia-18-112.
    [6] A. Holhoş, Uniform approximation by positive linear operators on noncompact intervals, Automat. Comput. Appl. Math., 18 (2009), 121-131. 
    [7] A. Holhoş, Uniform weighted approximation by positive linear operators, Stud. Univ. Babes-Bolyai Math, 56 (2011), 135-146. 
    [8] V. Mihesan, Uniform approximation with positive linear operators generated by generalized Baskakov method, Autmat. Comput. Appl. Math., 7 (1998), 38-97. 
    [9] A. Wafi and S. Khatoon, Direct and inverse theorems for generalized Baskakov operators in polynomial weight spaces, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 50 (2004), 159-173. 
    [10] A. Wafi and S. Khatoon, On the order of approximation of functions by generalized Baskakov operators, Indian J. Pure Appl. Math., 35 (2004), 347-358. 
    [11] A. Wafi and S. Khatoon, Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators, Cent. Eur. J. Math., 6 (2008), 325-334.  doi: 10.2478/s11533-008-0025-9.
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