Special Moduli of Continuity and the Constant in the Jackson-Stechkin Theorem

We consider a special 2k-order modulus of continuity W 2k(f,h) of 2π-periodic continuous functions and prove an analog of the Bernstein-Nikolsky-Stechkin inequality for trigonometric polynomials in terms of W 2k. We simplify the main construction from the paper by Foucart et al. (Constr. Approx. 29(2), 157-179, 2009) and give new upper estimates of the Jackson-Stechkin constants. The inequality W2k(f,h)≤3∥f∥∞ and the Bernstein-Nikolsky-Stechkin type estimate imply the Jackson-Stechkin theorem with nearly optimal constant for approximation by periodic splines. © 2013 The Author(s)