A sharp constant in the Jackson-Stechkin inequality in the space \(L^{2}\) on the period (Q643731)

Let \(E_n(f)\) denote the value of the best approximation in \(L^2(0, 2 \pi)\) of a function \(f\) by trigonometric polynomials of order at most \(n\), and let \(\omega(\delta, f)\) be the modulus of continuity of \(f\) of order \(1\) or \(2\). The author studies the sharp constant \(K_n(\delta, \omega)\) in the Jackson--Stechkin inequality \[ E_n(f)\leq K_n(\delta, \omega) \omega(\delta,f), \] where \[ K_n(\delta, \omega)= \sup \left \{ {{E_n(f)}\over{\omega(\delta,f)}}: f \in L^2(0, 2 \pi) \right \}, \] and finds the value of this constant for particular values of \(\delta\).