zbMATH — the first resource for mathematics

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 \[ \left(\frac1{2\pi}\int^{2\pi}_0\psi(x)dx\right)^{-1/2}. \]

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
Full Text: DOI