On upper bounds of Fourier–Walsh coefficients

An upper bound is established for the upper bounds of the Fourier–Walsh coefficients $a_n(f)$ whose modulus of continuity $\omega(\delta,f)$ does not exceed a given modulus of continuity $\omega(\delta)$. In the case of convex majorants of $\omega(\delta)$, these bounds are attained for individual ordinal numbers $n$.