Quantitative Estimates

Vijay Gupta, Themistocles M. Rassias, P. N. Agrawal and Ana Maria Acu
Additional contact information
Vijay Gupta: Netaji Subhas Institute of Technology
Themistocles M. Rassias: National Technical University of Athens
P. N. Agrawal: Indian Institute of Technology
Ana Maria Acu: Lucian Blaga University of Sibiu

Chapter Chapter 2 in Recent Advances in Constructive Approximation Theory, 2018, pp 29-72 from Springer

Abstract: Abstract The well-known theorem due to Bohman and Korovkin (cf. [62, 152]) states that if {L n} is a sequence of positive linear operators on the space C[a, b], then L n f → f for every f ∈ C[a, b], provided L n(e r, x) → e r(x), r = 0, 1, 2 with e r(t) = t r for n sufficiently large. Efforts have been made by several researchers to enlarge the domain of approximation operators and to include bounded or unbounded functions. A systematic study on Korovkin-type theorems was done by Altomare in [27], who provided applications concerning the approximation of continuous functions (as well as of L p-functions), by means of linear positive operators. Also, Boyanov and Veselinov [63] established the uniform convergence of any sequence of positive linear operators. Suppose C ∗[0, ∞) denotes the subspace of all real-valued continuous functions possessing a finite limit at infinity and equipped with the uniform norm. Boyanov and Veselinov proved the following theorem for the general sequence of linear positive operators:

Date: 2018
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DOI: 10.1007/978-3-319-92165-5_2

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