Sharp Constant in a Jackson-Type Inequality for Approximation by Positive Linear Operators

Abstract

In what follows, C is the space of $2{\pi }$-periodic continuous real-valued functions with uniform norm, $\omega \left( {f,h} \right) = \mathop {\sup }\limits_{\left| t \right| \leqslant h,x \in \mathbb{R}} \left| {f\left( {x + t} \right) - f\left( x \right)} \right|$ is the first continuity modulus of a function $f \in C$ with step h, H _n is the set of trigonometric polynomials of order at most n, $\mathcal{L}_n^ +$ is the set of linear positive operators $U_n :C \to H_n$ (i.e., of operators such that $U_n \left( f \right) \geqslant 0$ for every $\left( f \right) \geqslant 0$), $L_2 \left[ {0,1} \right]$ is the space of square-integrable functions on $\left[ {0,1} \right]$,

$${\lambda }_n \left( \gamma \right) = \mathop {\inf { sup}}\limits_{U_n \in \mathcal{L}_n^ + { }f \in C} \frac{{\left\| {f - U_n \left( f \right)} \right\|}}{{\omega \left( {f,\frac{{\gamma {\pi }}}{{n + 1}}} \right)}},{ \lambda }\left( \gamma \right) = \mathop {{sup}}\limits_{n \in \mathbb{Z}_ + } {\lambda }_n \left( \gamma \right).$$

It is proved that ${\lambda }_n \left( \gamma \right)$ coincides with the smallest eigenvalue of some matrix of order n+1. The main result of the paper states that, for every $\gamma >0,{\lambda }\left( \gamma \right)$ does not exceed and, for $\gamma \in \left( {0,1} \right]$, is equal to the minimum of the quadratic functional

$$\left( {B_\gamma \varphi ,\varphi } \right) = \frac{1}{{\pi }}\int\limits_0^\infty {\left( {1 + \left[ {\frac{t}{{\gamma {\pi }}}} \right]} \right)} \left| {\int\limits_0^1 {\varphi \left( x \right)e^{itx} dx} } \right|^2 dt$$

over the unit sphere of $L_2 \left[ {0,1} \right]$. Then it is calculated that ${\lambda }\left( {1} \right) = 1.312....$ Bibliography: 19 titles.