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

Abstract

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 $K(n,Q) \geqslant 1/\sqrt 2$ and $K(n,[0,\pi /\pi ]) = 1/\sqrt 2$. On the strength of a result of Yudin, we show that if the measure of the set Q is less than $\pi /n$, then $K(n,Q) >1/\sqrt 2$.