On certain properties of Rademacher chaos (Q2435812)

Let \(\{r_n\}_{n=0}^\infty\) be the Rademacher functions on \(\mathbb{R}\), namely, \(r_0(t)=1\) and \(r_n(t)=\text{sign}\,(\sin(2^n\pi t))\) for all \(t\in \mathbb{R}\) and \(n\in\mathbb{N}\). For \(d\in\mathbb{N}\), the set of all functions of the form \[ r_{i_1,\ldots,i_d}(t):=r_{i_1}(t)\cdots r_{i_d}(t),\quad t\in\mathbb{R}, \] with \(1\leq i_1<\cdots<i_d<\infty\) is called the Rademacher chaos of order \(d\). The paper is devoted to two properties of the Rademacher chaos: the \(\varepsilon\)-uniqueness and the strict convergence. For any \(\varepsilon>0\), a sequence \(\{f_n\}_{n=1}^\infty\) of functions on \([0,1]\) is an \(\varepsilon\)-uniqueness system if the convergence of the series \(\sum_{n=1}^\infty c_n f_n(t)\) to zero on an arbitrary set \(E\subset [0,1]\) satisfying \(m(E)>1-\varepsilon\) implies \(c_n=0\) for all \(n\in\mathbb{N}\). A system \(\{f_n\}_{n=1}^\infty\) of functions on \([0,1]\) is called a system of convergence if any series \(\sum_{k=1}^\infty c_kf_k\) converges a.e. on \([0,1]\) provided \(\{c_k\}_{k=1}^\infty\in \ell_2\). If this convergence a.e. on \([0,1]\) (respectively, on a set \(E\subset [0,1]\) with \(m(E)>0\)) implies that \(\{c_k\}_{k=1}^\infty\in \ell_2\), then \(\{f_n\}_{n=1}^\infty\) is called a system of semistrict convergence in the wide (narrow) sense. When \(\{f_n\}_{n=1}^\infty\) is both a system of convergence and a system of semistrict convergence in the wide (narrow) sense, then it is called a system of strict convergence in the wide (narrow) sense. In this paper, it is proved that the Rademacher chaos of order \(d\) is a system of \(2^{-d}\)-uniqueness, and a system of strict convergence in the wide sense but not in the narrow sense.