Sur une inégalité de Littlewood-Salem. (On a Littlewood-Salem inequality)
The random series \(\sum^\infty_{k= 1} X_k f(n_k x)\) is considered, where \(\{X_k\}\) is a sequence of independent complex random variables with \(EX_k= 0\) and \(E|X_k|^2< \infty\), \(k\geq 1\); \(f\) is a function in the class \(A(\mathbb{T})\), that is, \[ \|f\|_{A(\mathbb{T})}:= \sum_{n\in\mathbb{Z}} |\widehat f(n)|< \infty, \] and \(\{n_k\}\) is a strictly increasing sequence of positive integers. Let \(\sigma^2_N:= \sum^N_{k=1} E|X_k|^2\). Under a certain condition on the sequence \(\{X_k\}\), the following is proved \[ \int_{\Omega} \sup_{N\geq 1} \sup_{x\in\mathbb{T}} \Biggl|\sum^N_{k= 1} X_kf(n_k x)\Biggr|/\{\sigma^2_N\log n_N\}^{1/2} d\mathbb{P}< \infty. \] This inequality is applied to study the local regularities of random series of Rademacher type \(\sum a_k\varepsilon_k f(n_kx)\) and nonstationary Gaussian type \(\sum a_k g_kf(n_k x)\). The case when the \(n_k\) are real numbers is also studied.
scientific article; zbMATH DE number 1901038