Abstract
In this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.
Highlights
The classical Bohman–Korovkin theorem is one of the pivotal results of approximation theory and several convergence theorems known in literature employ this basic tool
It states that a sequence of positive linear operators Tnf acting on the set of the continuous functions over a compact interval of the real line converges to the identity operator only if it converges on a finite number of test functions which form a so-called Chebyshev system
If a sequence of operators Tnf is such that Tnφ = φ for some continuous function φ, to obtain the convergence appears very simple, if the functions φ belong to a Chebyshev system
Summary
The classical Bohman–Korovkin theorem (see [10,22,23]) is one of the pivotal results of approximation theory and several convergence theorems known in literature employ this basic tool It states that a sequence of positive linear operators Tnf acting on the set of the continuous functions over a compact interval of the real line converges to the identity operator only if it converges on a finite number of test functions which form a so-called Chebyshev system. A complete treatment of the Korovkin theorem can be found in the monographies [2,3] In this respect, if a sequence of operators Tnf is such that Tnφ = φ for some continuous function φ, to obtain the convergence appears very simple, if the functions φ belong to a Chebyshev system. That general approaches to convergence of integral operators was recently given in [9], and more recently in [20] and [25] in the frame of nonlinear operators
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