Mean Approximation of Functions on the Real Axis by Algebraic Polynomials with Chebyshev–Hermite Weight and Widths of Function Classes

S. B. Vakarchuk

Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 666-684

Résumé

We obtain sharp Jackson–Stechkin type inequalities on the sets $L^r_{2,\rho}(\mathbb{R})$ in which the values of best polynomial approximations are estimated from above via both the moduli of continuity of $m$th order and $K$-functionals of $r$th derivatives. For function classes defined by these characteristics, the exact values of various widths are calculated in the space $L_{2,\rho}(\mathbb{R})$. Also, for the classes $W^r_{2,\rho}(\mathbb{K}_m,\Psi)$, where $r=2,3,\dots$, the exact values of the best polynomial approximations of the intermediate derivatives $f^{(\nu)}$, $\nu=1,\dots,r-1$, are obtained in $L_{2,\rho}(\mathbb{R})$.

Détail

Keywords: mean approximation by algebraic polynomials, Jackson–Stechkin type inequalities, Chebyshev–Hermite weight, width of a function class, Fourier–Hermite series, modulus of continuity
Mots-clés : Hölder's inequality.

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