On the Fourier-Haar series of superpositions of functions (Q1385855)

Let \(f\in L[0,1], \{\phi_m\}_1^{\infty}\) the Haar system and \(\{a_m\}_1^{\infty}\) the sequence of Fourier-Haar coefficients. Assume that \(\omega(\delta)\) is a modulus of continuity and \(\{\tau(m)\}_1^{\infty}\) is a nonnegative sequence. For a function \(f\) we define the quantity \(A_{\omega,\tau}(f)=\sum_{m=2}^{\infty} \omega(| a_m(f)|)\tau(m)\). For almost increasing \(\tau(t)\) with the property \(\tau(2^{n+1})\leq C\tau(2^n+1), n\geq 0,\) the inequality \(A_{\omega,\tau}(\phi(f))\leq C(\omega,\tau)A_{\omega,\tau}(f)\) is fulfilled for all \(f\in L[0,1]\) and \(\phi\in \text{Lip}_D1\) if and only if the functions \(\omega\) and \(\tau\) satisfy the condition \[ \sum_{j=0}^N\tau(2^{N-j+1})\omega(\delta 2^{-j/2})\leq C\tau(2^{N+1})\omega(\delta). \] A similar result is true for \(\tau(m)\) with the property \(\tau(2^{n+1})\geq C\tau(2^n+1)\).