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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’skiĭ-type inequalities) 41A50 Best approximation, Chebyshev systems 42A10 Trigonometric approximation
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