On the Continuity of the Sharp Constant in the Jackson–Stechkin Inequality in the Space $L^2$
V. S. Balaganskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Abstract:
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.
Keywords:
approximation of a function, Jackson–Stechkin inequality, trigonometric polynomial, the space $L^2$, Tietze–Urysohn theorem, modulus of continuity, extremal function.
UDC:
517.5
Received: 10.06.2009
Revised: 23.03.2012
DOI:
10.4213/mzm8575
Fulltext:
PDF file (580 kB)
