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Direct theorems of trigonometric approximation for variable exponent Lebesgue spaces. (English) Zbl 1419.42001

Summary: Jackson type direct theorems are considered in variable exponent Lebesgue spaces \(L^{p(x)}\) with exponent \(p(x)\) satisfying \(1 \leq ess \inf_{x \in [0,2 \pi]} p(x), ess \sup_{x \in [0,2 \pi]} p(x) < \inf\), and the Dini-Lipschitz condition. Jackson type direct inequalities of trigonometric approximation are obtained for the modulus of smoothness based on one sided Steklov averages \[\mathfrak{Z}_v f(\cdot): = \frac{1}{v}\int^v_0 f(\cdot + t) dt\] in these spaces. We give the main properties of the modulus of smoothness \[\Omega_r(f,v)_{p(\cdot)}:= ||(\mathbf{I}- \mathfrak{Z}_v)^rf||_{p(\cdot} \quad (r \in \mathbb{N}) \]in \(L^{p(x)}\) where \(\mathbf{I}\) is the identity operator. An equivalence of the modulus of smoothness and Peetre’s \(K\)-functional is established.

MSC:

42A10 Trigonometric approximation
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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