On the Continuity of the Sharp Constant in the Jackson--Stechkin Inequality in the Space~$L^2$

V. S. Balaganskii

Matematičeskie zametki, Tome 93 (2013) no. 1, pp. 13-28

Résumé

This paper deals with the continuity of the sharp constant $K(T,X)$ with respect to the set $T$ in the Jackson–Stechkin inequality $$ E(f,L)\le K(T,X)\omega(f,T,X), $$ where $E(f,L)$ is the best approximation of the function $f\in X$ by elements of the subspace $L\subset X$, and $\omega$ is a modulus of continuity, in the case where the space $L^2(\mathbb T^d,\mathbb C)$ is taken for $X$ and the subspace of functions $g\in L^2(\mathbb T^d,\mathbb C)$, for $L$. In particular, it is proved that the sharp constant in the Jackson–Stechkin inequality is continuous in the case where $L$ is the space of trigonometric polynomials of $n$th order and the modulus of continuity $\omega$ is the classical modulus of continuity of $r$th order.

Détail

Keywords: approximation of a function, Jackson–Stechkin inequality, trigonometric polynomial, the space $L^2$, Tietze–Urysohn theorem, modulus of continuity, extremal function.