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Extension technique and estimates for moduli of smoothness on domains in \(\mathbb{R}^d\). (English) Zbl 1234.41008

The present paper deals with the extension technique which allows to deduce inequalities involving regular moduli of smoothness on a bounded domain in \(\mathbb R^d\) from inequalities involving moduli of smoothness of periodic functions on \(T^d=[-\pi,\pi]^d\). The goal is to apply the extension technique to establish sharp lower estimates for moduli of smoothness on bounded Lipschitz-graph domains in \(\mathbb R^d\) for rearrangement-invariant Banach function spaces satisfying certain conditions. These inequalities give an significant improvement over the classical immediate estimate \(\omega^{r+1}(f,t)_B\leq 2\omega^r(f,t)_B\).

MSC:

41A25 Rate of convergence, degree of approximation
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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