×

zbMATH — the first resource for mathematics

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’skiń≠-type inequalities)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] N. I. Chernykh, ”On the best approximation of periodic functions by trigonometric polynomials inL 2”,Mat. Zametki [Math. Notes],2, No. 2, 513–522 (1967).
[2] L. V. Taikov, ”The best approximation of differentiable functions in the metric of the spaceL 2,”Mat. Zametki [Math. Notes],2, No. 4, 535–542 (1977).
[3] L. V. Taikov, ”Structural and constructive characteristics of functions fromL 2,”Mat. Zametki [Math. Notes],25, No. 2, 217–223 (1979).
[4] N. Ainullaev, ”The best approximation of some classes of differentiable functions inL 2”, in:Application of Functional Analysis in Approximation Theory. Collection of Scientific Papers, [in Russian], Kalinin University, Kalinin (1991).
[5] V. V. Shalaev, ”On the widths inL 2 of classes of functions defined by the moduli of continuity of high order”,Ukrain. Mat. Zh. [Ukrainian Math. J.],43, No. 3, 125–129 (1991).
[6] M. G. Esmaganbetov, ”On the widths inL 2 of classes of differentiable functions,”, in:International Conference ”Function Spaces, Approximation Theory, Nonlinear Analysis”, Dedicated to the 90th anniversary of S. M. Nikol’skii [in Russian], Abstracts of Papers, Moscow (1995), pp. 124–125.
[7] N. P. Korneichuk,Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.