The sharp constant in Jackson-type inequality for approximation by linear positive operators
O. L. Vinogradov St. Petersburg State University, Department of Mathematics and Mechanics
Abstract:
In what follows,
$C$ is the space of
$2\pi$-periodic continuous real-valued functions with uniform norm,
$\omega(f,h)=\sup_{|t|\le{h},x\in\mathbb R}|f(x+t)-f(x)|$ is the first modulus of continuity of function
$f\in C$ with step
$h$,
$H_n$ is the set of trigonometric polynomials of order not greater than
$n$,
${\mathscr L}_n^+$ is the set of linear positive operators
$U_n:C\to H_n$ (i.e. such that
$U_n(f)\ge0$ for every
$f\ge0$),
$L_2[0,1]$ is the space of square integrable on
$[0,1]$ functions,
$$
\lambda_n(\gamma)=\inf_{U_n\in{\mathscr L}_n^+}\sup_{f\in C}\frac{\|f-U_n(f)\|}{\omega(f,\frac{\gamma\pi}{n+1}}, \qquad \lambda(\gamma)=\sup_{n\in\mathbb Z_+}\lambda_n(\gamma).
$$
It is proved that
$\lambda_n(\gamma)$ coincides with the smallest eigenvalue of some matrix of order
$n+1$. The principal result of the paper is the following: for every
$\gamma>0$ $\lambda(\gamma)$ doesn't outnumber and for
$\gamma\in(0,1]$ is equal to the minimum of square functional
$$
(B_{\gamma}\varphi,\varphi)=\frac1\pi\int\limits_0^{\infty}\biggl(1+\biggl[\frac{t}{\gamma\pi}\biggr]\biggr)\Biggl|\int\limits_0^1\varphi(x)e^{itx}\,dx\Biggr|^2dt
$$
on the unit sphere of
$L_2[0,1]$. Then it is calculated that
$\lambda(1)=1.312\ldots$
UDC:
517.5
Received: 17.03.1997