A theorem related to Marcinkiewicz-Salem conjecture
We recall that the Riemann means \(R_nf\) of a measurable function \(f\) on the interval \([0,1)\) are defined by \[ R_nf(x):= {1\over n} \sum^{n-1}_{k=0} f\Biggl(x+{k\over n}\Biggr),\quad n\in\mathbb{N}. \] If \(f\in L^2[0,1)\) with Fourier expansion of the form \[ \sum_{j\in\mathbb{N}} a_j e_j,\quad\text{where }e_j(x):= \exp(2i\pi jx), \] then \[ R_nf(x)= \sum_{j\in\mathbb{N},n| j}a_j e_j. \] The present author considers the randomly sampled trigonometric system \(\{e_{S_j}:j\in \mathbb{N}\}\), where \(S_j:= \varepsilon_1+ \varepsilon_2+\cdots+ \varepsilon_j\) and \(\{e_j:j\in \mathbb{N}\}\) is a Bernoulli sequence: \(P\{\varepsilon_j= 0\}= P\{\varepsilon_j= 1\}= 1/2\) with \((\Omega,{\mathcal A},P)\) as basic probability space. Three theorems are proved. The following is typical. Theorem 1. For any \(\{a_j:j\in \mathbb{N}\}\in \ell^2(\mathbb{N})\) and \(\widetilde f= \sum_{j\in\mathbb{N}} a_j e_{S_j}\) we have \[ \lim_{m\to\infty}\, {1\over m}\,\sum^m_{n=1} R_n\widetilde f= 0\qquad\text{a.s.,} \] where the abbreviation a.s. means that the convergence holds for almost all \(x\in [0,1)\) and almost all \(\omega\) in \(\Omega\).
scientific article; zbMATH DE number 2111453