On the stability of some classical operators from approximation theory. (English) Zbl 1284.39032
An operator \(T\) mapping a normed space \(A\) into a normed space \(B\) is Hyers-Ulam stable (HU-stable) iff there exists a constant \(K\) such that for any \(g\in T(A)\), \(\varepsilon>0\) and \(f\in A\) such that \(\|Tf-g\|\leq\varepsilon\), there exists \(f_0\in A\) such that \(Tf_0=g\) and \(\|f-f_0\|\leq K\varepsilon\). The infimum of all constants \(K\) is denoted by \(K_T\).
The paper deals with HU-stability of several classical operators from approximation theory. In particular, the stability of Bernstein operators \(B_n\) is proved and the constant \(K_{B_n}\) is determined. For the Szász-Mirakjan operators \(L_n\) the lack of stability is shown. The other considered operators are: the Meyer-König and Zeller operators, Kantorovich, Durmeyer, Bernstein-Durmeyer and beta integral operators, projections and others.
The paper deals with HU-stability of several classical operators from approximation theory. In particular, the stability of Bernstein operators \(B_n\) is proved and the constant \(K_{B_n}\) is determined. For the Szász-Mirakjan operators \(L_n\) the lack of stability is shown. The other considered operators are: the Meyer-König and Zeller operators, Kantorovich, Durmeyer, Bernstein-Durmeyer and beta integral operators, projections and others.
Reviewer: Jacek Chmieliński (Kraków)
MSC:
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 41A35 | Approximation by operators (in particular, by integral operators) |
| 41A44 | Best constants in approximation theory |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
Keywords:
Hyers-Ulam stability; approximation; best constant; Meyer-König operator; Kantorovich operator; Durmeyer operator; normed space; Bernstein operator; Szász-Mirakjan operator; beta integral operatorReferences:
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