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Widths of classes from $$L_2[0,2\pi]$$ and minimization of exact constants in Jackson-type inequalities. (English. Russian original) Zbl 0960.42002
Math. Notes 65, No. 6, 689-693 (1999); translation from Mat. Zametki 65, No. 6, 816-820 (1999).
In the paper the problem of minimizing the constant in Jackson-type inequalities for the space $$L^2$$ of $$2\pi$$-periodic functions with respect to all subspaces $$K_N^T$$ of dimension $$N$$ is solved. The norm in $$L^2$$ is defined by the equality $$\|f\|=({1\over \pi}\int_{0}^{2\pi}|f(x)|^{2} dx)^{1/2}$$. By $$L_2^{\alpha}$$ we denote the set of functions $$f\in L^2$$ for which there exists the Weyl derivative $$f^{(\alpha)}\in L^2$$ ($$\alpha\geq 0$$). For a function $$f\in L^2$$ we define the modulus of smoothness of order $$r\geq 0$$ by the equality $\omega_{r}(f,\delta)=\sup_{0\leq h\leq\delta}\left\|\sum_{k=0}^r (-1)^k {r\choose k}f(x+(r-k)h)\right\|.$
For $$\delta,\vartheta>0$$, $$\alpha,r,\beta,\gamma\geq 0$$ we set $W_r (f^{(\alpha)},\delta)=\left( \int_0^\delta \omega_r^\vartheta(f^{(\alpha)};t)\sin^\gamma (\beta t/\delta) dt \over \int_0^\delta \sin^\gamma (\beta t/\delta) dt \right)^{1/\vartheta}$ and $X_{N,\delta,r,\beta,\gamma,\vartheta}(L_2^\alpha,L_2)= \inf_{K_N^T}\sup_{f\in L^2} {E(f;K_N^T) \over W_r (f^{(\alpha)};\delta)_{\beta,\gamma,\vartheta}}$ where $$E(f;K_N^T)=\inf\{\|f-g\|:g\in K_N^T\}$$. One part of Theorem 2 states that for all $$r\geq 0$$, $$\alpha\geq 1$$, $$0<\vartheta\leq 2$$, $$0<\delta< {\pi\over n}$$, $$0\leq\gamma\leq\vartheta\alpha-1$$, $$0<\beta\leq\pi$$, $$N=2n$$ or $$N=2n-1$$ the equality $X_{N,\delta,r,\beta,\gamma,\vartheta}(L_2^\alpha,L_2)={1\over n^\alpha} \left( \int_0^\delta \sin^\gamma (\beta t/\delta) dt \over \int_0^\delta (2\sin(nt/2))^{r\vartheta}\sin^\gamma (\beta t/\delta) dt \right)^{1/\vartheta} \tag $$*$$$ is valid.
The other part states that the quantity $$(*)$$ equals to $$N$$-dimensional Kolmogorov width, linear width and projection width of the class $$L_2^\alpha(\delta,r,\beta,\gamma,\vartheta)= \{f\in L_2^\alpha:W_{r}(f^{(\alpha)};\delta)\}_{\beta,\gamma,\vartheta}$$ in $$L_2$$.

##### MSC:
 42A10 Trigonometric approximation 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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##### References:
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