Trigonometric Fourier series and Walsh-Fourier series with lacunary sequence of partial sums
Let \(\psi\) be a trigonometric or Walsh system on the set \([0,1)\). Let \(k\in\mathbb N\) be a fixed number, \(J_k=\{j_1,j_2,\dots,j_k\}\) be a fixed subset of \(\{1,2,\dots,d\}\), \(\mathbb N_{j_\nu}\) be a lacunary sequence, and \[ \mathbb N^d(J_k)=\mathbb N\times\dots\times\mathbb N\times\mathbb N_{j_1}\times\mathbb N\times\dots\times\mathbb N\times\mathbb N_{j_k}\times\mathbb N\times\dots\times\mathbb N. \] Let \(E\subset[0,1)^d\) and \(\mu E>0\). We say that, for multiple Fourier series with respect to the system \(\psi\) in the class \(L_p([0,1)^d)\), \(p>1\), weak generalized localization almost everywhere takes place on the set \(E\subset[0,1)^d\) with respect to the set \(\mathbb N^d(J_k)\), if for any function \(f\in L_p([0,1)^d)\), \(f(x)=0\) on \(E\), there exists a subset \(E_1\subset E\), \(\mu E_1>0\) such that \[ \lim\limits_{{n\to\infty\atop n\in\mathbb N^d(J_k)}}S_n(f,x,\psi)=0 \] almost everywhere on \(E_1\). The authors find a necessary and sufficient condition on the set \(E\), under which for multiple Fourier series with respect to the system \(\psi\) in the class \(L_p([0,1)^d)\) weak generalized localization takes place on the set \(E\subset[0,1)^d\) with respect to the set \(\mathbb N^d(J_k)\).
scientific article; zbMATH DE number 6158191
scientific article