Optimal Set of the Modulus of Continuity in the Sharp Jackson Inequality in the Space $L_2$

E. E. Berdysheva

Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 666-674

Résumé

To a function $f\in L_2[-\pi,\pi]$ and a compact set $Q\subset[-\pi,\pi]$ we assign the supremum $\omega(f,Q) =\sup_{t\in Q}\|f(\,\cdot\,+t)-f(\,\cdot\,)\|_{L_2[-\pi,\pi]}$, which is an analog of the modulus of continuity. We denote by $K(n,Q)$ the least constant in Jackson's inequality between the best approximation of the function $f$ by trigonometric polynomials of degree $n-1$ in the space $L_2[-\pi,\pi]$ and the modulus of continuity $\omega(f,Q)$. It follows from results due to Chernykh that $K(n,Q)\ge1/\sqrt2$ and $K(n,[0,\pi/n])=1/\sqrt2$. On the strength of a result of Yudin, we show that if the measure of the set $Q$ is less than $\pi/n$, then $K(n,Q)>1/\sqrt2$.