Direct and converse theorems of Jackson type in $L^p$ spaces, $0<p<1$

In this paper the connection between properties of functions and best approximations in classical orthogonal systems is studied in the $L^p$-metric, $0<p<1$. Two-sided inequalities are established between moduli of continuity and best approximations in these systems, which are unimprovable in a well-defined sense. Inequalities between best approximations in various metrics are also presented. A number of results are generalized to the classes $\varphi(L)$.
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