Ukrainian Mathematical Bulletin

Volume 3, Issue , 001 2006 pp. 113-130.

On Jeckson Type Inequalities in $L_2[-1,1]$ and Exact Values of $n$-widths of Functional Classes

Authors: S. B. Vakarchuk
Author institution: Ukrainian Custom Academy, ul. Dzerzhynskogo, 2/4, 49044, Dnepropetrovsk, Ukraine, academy@amsu.dnp.ukrpack.net

Summary: For the Hilbert space $L_2[-1,1]$, we find two kinds of exact Jeckson type inequalities. One of them gives a relation between the quantity $E_{n-1}(f)$, which is the best approximation of a non-periodic function $f$ with a subspace of algebraic polynomials of degree $(n-1)$, and the generalized continuity modulus of order $s$, $\omega_{s}^{L}(\mathcal{D}^{r}f, \tau)$, introduced by P.~Butser for the strong Legendre derivative $\mathcal{D}^r f$ of order $r$. Another inequality establishes a relation between $E_{n-1}(f)$ and the $K$-functional $K_S (\mathcal{D}^r f,\tau)$. For the classes $W^{r}\!(K_s,\!\Psi)$, $r,\ s\in \mathbb{N}$, which are defined in terms of the $K$-functional $K_s (f,\!\tau)$ and the majorant $\Psi(\tau)$, of functions that have the derivatives $\mathcal{D}^rf$ satisfying the condition $K_s(\mathcal{D}^rf, \tau)\leqslant\Psi(\tau)$ for any $0\tau\leqslant2$, we find exact values of some $n$-widths. For these classes, we find exact upper bounds for absolute values of the coefficients in the decomposition of the function into a series in Legendre polynomials.

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