Optimal Set of the Modulus of Continuity in the Sharp Jackson Inequality in the Space L2

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 \(K(n,Q) >1/\sqrt 2 \).