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On Jackson’s inequality in \(L_2\) with a generalized modulus of continuity. (English. Russian original) Zbl 1068.41026
Sb. Math. 195, No. 8, 1073-1115 (2004); translation from Mat. Sb. 195, No. 8, 3-46 (2004).
The authors study the sharp constant \(\kappa = \kappa(\Lambda,T)\) in the Jackson type inequality in the space \(L_2({\mathbb T}^d)\) \[ E_{\Lambda}(f) \leq \kappa \overline{\omega}_{\psi}(f,T), \quad f \in L_2({\mathbb T}^d). \] Here \(E_{\Lambda}(f)\) is the best approximation of \(f\) in \(L_2({\mathbb T}^d)\) by functions with spectrum in a set \(\Lambda \subset {\mathbb Z}^d\), \(T\) is a closed subset of \({\mathbb T}^d\) and \[ \overline{\omega}_{\psi}(f,T) = \max_{t \in T}{\sqrt{\sum_{s \in {\mathbb Z}^d} \psi(st)| \widehat f_s| ^2} } \] is the generalized modulus of continuity. In the last definition \(\psi\) is a non-negative continuous \(2\pi\)-periodic function. The minimum constant in the Jackson inequality is \[ \kappa^*(\Lambda) = \min{\{ \kappa(\Lambda,T) : T \;\text{is a closed subset of} \;{\mathbb T}^d \}}. \]
The value of \(\kappa^*(\Lambda)\) is found under quite general conditions on \(\Lambda\) and \(\psi\). This result contains and sometimes improves a number of known results.
In the one-dimensional case, a special class of generalized moduli is introduced which contains the moduli \(\tilde{\omega}_{a,r}(f,\delta) = \sup_{0 \leq t \leq \delta}{\| \Delta_{a^{r-1}t} \cdots \Delta_{at} \Delta_t f\| _2}\), with \(a\) even. The relation \(\kappa(\{-n+1,n-1\},[0,\delta]) = \kappa^*(\{-n+1,n-1\})\) is proved in this class for all \(\delta \geq \frac{\pi}{n}\).

MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
41A50 Best approximation, Chebyshev systems
42A10 Trigonometric approximation
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