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Direct theorems of trigonometric approximation for variable exponent Lebesgue spaces
Volume 60, no. 1 (2019), pp. 121–135

Abstract

Jackson type direct theorems are considered in variable exponent Lebesgue spaces $L^{p(x)}$ with exponent $p(x)$ satisfying $1\leq \operatorname{ess\,inf}_{x\in [0,2\pi ]}p(x)$, $\operatorname{ess\,sup}_{x\in [0,2\pi]}p(x) < \infty$, 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 $\mathfrak{Z}_{v}f(\cdot) := \frac{1}{v}\int\nolimits_{0}^{v}f (\cdot+t) \,dt$ in these spaces. We give the main properties of the modulus of smoothness $\Omega_{r}(f,v)_{p(\cdot)} := \left\Vert (\mathbf{I}-\mathfrak{Z}_{v})^{r}f\right\Vert_{p(\cdot)}\quad (r\in \mathbb{N})$ 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.