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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 W 2k (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.

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