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Inequalities for Moduli of Continuity in Abstract Banach Spaces

Authors:	V V Zhuk G I Natanson

Abstract:	Suppose that X is a Banach space, K denotes the set of real numbers R or the set of nonnegative real numbers R _{+}, $A\left( K \right) = \left\{ {T_{{\alpha }} } \right\}{\alpha }{\kern 1pt} \in {\kern 1pt} {K}$ is a family of linear operators from X into X such that T _0=I is the identity operator in X, $T_{\alpha } T_{\beta }$ for all ${\alpha ,\beta } \in K$ , and there exists M such that $\left\\| {T_{\alpha } \left( x \right)} \right\\| \leqslant M\left\\| x \right\\|$ for all $\alpha \in {\text{K,}}x \in X$ . The expression $\eta _r \left( {x,h} \right) = \mathop {\sup }\limits_{T_t \in A\left( K \right),\left\| t \right\| \leqslant h} \left\\| {\left( {T_t - I} \right)^r \left( x \right)} \right\\|$ is called the rth order modulus of continuity of an element x with step h in the space X with respect to the family A(K). The properties of $\eta _r \left( {x,h} \right)$ are studied. Bibliography: 3 titles.

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