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Some properties of moduli of smoothness of sets in the space of continuous functions. (English) Zbl 0588.41004

Let X be a nonempty and bounded subset of the space \(C<a,b>\) and let \(x\in X\). For a given \(\epsilon >0\) and a positive integer n we denote \(\Delta^ n_ hx(t)=\sum^{n}_{j=0}(-1)^{n-j}\left( \begin{matrix} n\\ j\end{matrix} \right)x(t+jh),\) \(t,t+h,...,t+nh\in <a,b>\), \(\omega^ n(x,\epsilon)=\sup [| \Delta^ n_ hx(t)|:\quad t+jh\in <a,b>,\quad j=0,1,...,n,\quad | h| \leq \epsilon],\quad \omega^ n(X,\epsilon)=\sup [\omega^ n(x,\epsilon):\quad x\in X].\) The function \(\epsilon \to \omega^ n(X,\epsilon)\) is called the modulus of smoothness of a set X of order n. In the paper many properties concerning the modulus \(\omega^ n(X,\epsilon)\) and its generalizations are proved. For example, a connection between \(\omega^ n(X,\epsilon)\) and the Hausdorff distance of a set X from the family of all polynomials of degree at most n-1 is given. The results obtained belong rather to approximation theory but some of them are expressed in terms of measures of noncompactness.

MSC:

41A10 Approximation by polynomials
46B20 Geometry and structure of normed linear spaces
47H10 Fixed-point theorems
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