Integrability of the sum of absolute values of blocks of the Fourier-Walsh series for functions of bounded variation
Suppose that \(n\in\mathbb Z_+=\{0,1,\dots\}\) has the binary expansion \(n=\sum^\infty_{k=0}\nu_k2^k\), \(\nu_k\in \{0,1\}\). Every \(x\in [0,1)\) may be represented in the form \(x=\sum^\infty_{k=0}x_k/2^{k+1}\), \(x_k\in\{0,1\}\). For \(x=l/2^n\), \(n\in\mathbb N\), \(l\in (0,2^n)\cap \mathbb Z\), we consider the expansion with a finite number of \(x_k\neq 0\).) Then the \(n\)-th Walsh function is defined by \[ w_n(x)=(-1)^{\sum^\infty_{k=0}\nu_kx_k}, \quad n\in\mathbb Z_+. \] The variation of \(n\in\mathbb Z_+\) is \(V(n)=\sum^\infty_{k=1}|\nu_k-\nu_{k-1}|+\nu_0\). If \(n\) and \(m\) have binary expansions \[ n=2^{l_1}+\dots+2^{l_j}+2^{l_{j+1}}+\dots 2^{l_s}, \quad m= 2^{k_1}+\dots+2^{k_i}+2^{l_{j+1}}+\dots 2^{l_s}, \] where \(l_1<l_2<\dots<l_s\), \(k_1<\dots<k_i<l_{j+1}<\dots <l_s\) and \(l_j\neq k_i\), then we set \(\widetilde{n}=2^{l_1}+\dots+2^{l_j}\) and \(\widetilde{m}=2^{k_1}+\dots+2^{k_i}\). Finally, \(V(n,m)=V(\widetilde{n})+V(\widetilde{m})\). The main result of this paper is Theorem 1. Let \(\{n_j\}^\infty_{j=1}\) be a strictly increasing sequence of positive integers. The sum \(\sum^\infty_{j=1}\left|\sum^{n_{j+1}-1}_{n=n_j}c_nw_n(x)\right|\), where \(c_n=c_n(f)\) are Walsh-Fourier coefficients of a function \(f\) of bounded variation on \([0,1]\), belongs to the Lebesgue space \(L^1[0,1)\) if and only if the series \(\sum^\infty_{j=1}V(n_j,n_{j+1})/n_{j+1}\) converges.
scientific article; zbMATH DE number 6527095