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On Jackson’s inequality with a generalized modulus of continuity. (English. Russian original) Zbl 1062.41009
Math. Notes 73, No. 5, 736-741 (2003); translation from Mat. Zametki 73, No. 5, 783-788 (2003).
Given a nonnegative $$2\pi$$-periodic continuous function $$\psi$$, $$\psi(0)=0$$, one defines the generalized modulus of continuity as $\bar{\omega}_{\psi}(f,T)=\max_{t\in T}\left(\sum_s\psi(st)| \hat{f}_s| ^2\right)^{1/2},$ where $$T$$ is a closed subset of $$[0,2\pi]^d$$. Let $$\Lambda$$ be a proper subset of $$\mathbb Z^d$$ that contains the origin and let $$E_{\Lambda}(f)$$ denote the error of best approximation to $$f\in L_2(\mathbb T^d)$$ by functions whose spectrum is concentrated on $$\Lambda$$. The authors show that the minimal sharp constant $$C$$ in Jackson’s inequality $E_{\Lambda}(f)\leq C\bar{\omega}_{\psi}(f,T)\quad\text{for each}\;f\in L_2(\mathbb T^d)$ is equal to $\left(\frac1{2\pi}\int^{2\pi}_0\psi(x)dx\right)^{-1/2}.$

##### MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
##### Keywords:
modulus of continuity; Jackson’s inequality
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