Abstract
In this paper, we construct a new variant of the classical Szász–Mirakyan operators, Mn, which fixes the functions 1 and eax,x≥0,a∈R. For these operators, we provide a quantitative Voronovskaya-type result. The uniform weighted convergence of Mn and a direct quantitative estimate are obtained. The symmetry of the properties of the classical Szász–Mirakyan operator and of the properties of the new sequence is investigated. Our results improve and extend similar ones on this topic, established in the last decade by many authors.
Highlights
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A lot of papers devoted to modifications of certain positive linear operators (p.l.o), which fix certain exponential functions, have been published
Apart from preservation of monomials, there is an increasing interest to modify the operators of Bernstein, Szász–Mirakyan, Baskakov, Phillips and their Kantorovich and Durrmeyer variants, such that the new modified operators reproduce certain exponential type-functions
Summary
A lot of papers devoted to modifications of certain positive linear operators (p.l.o), which fix certain exponential functions, have been published. King in his famous paper [1] from 2003, where he modified the Bernstein operator to achieve better approximation on some subintervals of [0, 1]. Later, this method was extended in the papers of Raşa, Aldaz, Kounchev, Render (see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]) and many others. Motivated by the results from [26], in this paper, we introduce operators which preserve the functions 1 and e ax for all a ∈ R \ {0}.
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