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Greediness of the wavelet system in \(L^{p(t)}(\mathbb{R})\) spaces. (English) Zbl 1217.41014

Summary: We point out that when Hardy-Littlewood maximal operator is bounded on the space \(L^{p(t)}(\mathbb R)\) (\(1<a\leq p(t)\leq b<\infty\); \(t\in\mathbb R\)), well-known characterization of \(L^p(\mathbb R)\) spaces (\(1<p<\infty\)) in terms of orthogonal wavelet bases extends to space \(L^{p(t)}(\mathbb R)\). As an application of this result, we show that such bases normalized in \(L^{p(t)}(\mathbb R)\) are not greedy bases of \(L^{p(t)}(\mathbb R)\) if \(p(t) \neq \text{const}\).

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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