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Strong converse inequalities. (English) Zbl 0798.41009

Let \(X\) be a linear normed space and \(Y\) a dense subspace induced by a given operator \(D\) by \(Y= \{f\in X\): \(Df\in X\}\). Then the Peetre \(K\)- functional is defined by \(f\in X\) by \(K(f,t)=\inf_{g\in Y} (\| f- g\|_ X+ t \| Dg\|_ X)\), \(t>0\). The \(K\)-functional is often equivalent to the rate of approximation of a given sequence of operators. The authors study the sequence of bounded operators (1) \(Q_ n: X\to X\) with \(\| Q_ n\|\leq M\) and the inequality (2) \(\| f-Q_ n f\|\leq C \lambda(n)\| Df\|\) for \(f\in Y\) is considered with \(\lambda(n)\downarrow 0\), \(C\) does not depend on \(f\), \(n\). The inequalities (2) are called the Jackson-type inequalities and they together with (1) give (3) \(\| f-Q_ n f\|\leq C_ 1 K(f,\lambda(n))\) (similar inequalities are considered if we have a family of operators). The inequalities (3) are called the direct estimate for the approximation process. They are interesting mainly when they are best possible in a certain sense. This leads to the study of strong converse inequalities. The authors define fourth types of such inequalities, describe relations between these types of inequalities and discuss the hierarchy of strong converse inequalities.

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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