Estimates of Functions, Orthogonal to Piecewise Constant Functions, in Terms of the Second Modulus of Continuity

Abstract

The paper is devoted to the problem of finding the exact constant \( {W}_2^{\ast } \) in the inequality ‖f‖ ≤ K ⋅ ω₂(f, 1) for bounded functions f with the property

\( \underset{k}{\overset{k+1}{\int }}f(x) dx=0,\kern0.5em k\in \mathrm{\mathbb{Z}}. \)

Our approach allows us to reduce the known range for the desired constant as well as the set of functions involved in the extremal problem for finding the constant in question. It is shown that \( {W}_2^{\ast } \) also turns out to be the exact constant in a related Jackson–Stechkin type inequality.