Estimation of functions orthogonal to piecewise constant functions in terms of the second modulus of continuity

The article is concerned with the question about the exact constant $W_2^*$ in the inequality $\|f\|\le K\cdot\omega_2(f,\,1)$ for bounded functions $f$ with the property
$$ \int_k^{k+1}f(x)\,dx=0,\qquad k\in\mathbb Z. $$

The approach suggested made it possible 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 $W_2^*$ also turns out to be the exact constant in a related Jackson–Stechkin type inequality.