Estimates for sums of moduli of blocks in trigonometric Fourier series (Q643851)

For a sequence \(A=\{A_k\}\), where \(A_k\) is a finite subset of \(\mathbb Z\), the author first studies the conditions on \(A\) and \(\Lambda = \{\lambda_k\}\), where \(\lambda_k: A_k \to \mathbb C\) is a sequence of functions, which guarantee the existence of a number \(C\) such that \(\|U_{A,\Lambda} (f)\|_p \leq C\|f\|_1\) for all \(f\in L_1\), where \(U_{A,\Lambda} (f)(x)=\sum_{k=1}^\infty |\sum_{m\in A_k}\lambda_k (m)c_m (f)e^{imx}|\), and \(c_m (f)\) are Fourier coefficients of the function \(f\). Then, he obtains conditions on \(A\) which guarantee that the function \(\sum_{k=1}^\infty|\sum _{m\in A_k}c_m(h)e^{imx}|\) belongs to \(L _{p}\) for a function of bounded variation.