Estimate for the difference of operators having different basis functions
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Estimate for the difference of operators having different basis functions Vijay Gupta1 Received: 10 August 2019 / Accepted: 30 August 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract Very recently Aral et al. (Anal Math Phys, 2018. https://doi.org/10.1007/s1332) established an estimate for the difference of two linear positive operators, but their result is valid for operators having same basis functions. In the present article, we study the approximation of difference of operators and establish a general estimate for the difference of those operators, which have different basis functions. In the end, we give examples of quantitative estimates for the difference between the Baskakov and Szász–Mirakyan operators and between Szász– Mirakyan and Baskakov–Kantorovich operators in terms of weighted modulus of continuity. Also, graphical representation is presented for such examples. Keywords Difference of operators · Szász–Mirakyan operators · Baskakov operators · Baskakov–Kantorovich operators · Weighted modulus of continuity Mathematics Subject Classification 41A25 · 41A35
1 Introduction After the work on difference of two operators by Acu and Rasa [1], several papers appeared in the literature on such topic for instance we mention here the recent works [2,4–8] etc. Aral et al. [3] considered such problem in terms of weighted modulus of continuity. Almost all work was done for two operators having same basis function. Now the natural question arise: “Can we consider the difference of two operators having different basis functions”, here we handle this problem and establish the difference of any two operators, even if they have different basis functions. n and Let us consider the two linear positive operators M L n as follows: n ( f , x) = M
Fkn ( f ) pn,k (x)
(1)
k∈K
B 1
Vijay Gupta [email protected] Department of Mathematics, Netaji Subhas University of Technology, Sector 3 Dwarka, New Delhi 110078, India
123
V. Gupta
and L n ( f , x) =
G nk ( f )vn,k (x)
(2)
k∈K
where K be a set of non-negative integers. By C2 [0, ∞) we mean the class of all continuous functions on positive real axis and f (x) = O(1 + x 2 ). Ispir [9], considered the following weighted modulus of continuity: ( f , δ) =
sup
| f (x + a) − f (x)|[(1 + x 2 )(1 + a 2 )]−1 .
x≥0,|a|
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