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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 \(L_2\).

42A10 Trigonometric approximation
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
Full Text: DOI
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