Optimal set of the modulus of continuity in the sharp Jackson inequality in the space L2

To a function f is an element of L-2[-pi, pi] and a compact set Q subset of [-pi, pi] we assign the supremum w(f, Q) = sup(tis an element ofQ) parallel tof( (.) + t) - f ((.))parallel toL(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 w(f, Q). It follows from results due to Chernykh that K(n, Q) greater than or equal to 1/root2 and K(n, [0, pi/n]) = 1/root2. 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/root2.