论文标题
在广义lac级系列中
On generalized lacunary series
论文作者
论文摘要
Given lacunary sequence of integers, $n_k$, $n_{k+1}/n_k>λ>1$, we define a new sequence $\{m_k\}$ formed by all possible $l$-wise sums $\pm n_{k_1}\pm n_{k_2}\pm \ldots\pm n_{k_l}$.我们证明如果$λ>λ_l$，则任何系列\ begin {equation} \ sum_kc_ke^{im_kx}，\ qquad（1）\ end {equation}带有$ \ sum_k | c_k | c_k |^2 <\ infty $在任何术语重新排列后几乎在任何地方收敛，其中$ 1 <λ_l<2 $是一定的关键值。我们建立了此属性，证明了一种新的Khintchine类型不等式$ \ | s \ | _p \ le c_ {l，λ，p} \ | s \ | _2 $，$ p> 2 $，其中$ s $是表格的有限总和（1）。 对于$λ\ ge 3 $，我们还为常数$ c_ {l，λ，p} $的生长建立了一个尖锐的费率$ p^{l/2} $作为$ p \ to \ to \ infty $。 Bonami和Kiener对Rademacher混乱总和进行了这种估计。对于$λ\ ge 3 $，我们还建立了一些系列（1）：1）的逆收敛属性，如果串联（1）收敛a.e.，则$ \ sum_k | c_k | c_k |^2 <\ infty $，2）如果a.e.收敛到零，然后$ c_k = 0 $。
Given lacunary sequence of integers, $n_k$, $n_{k+1}/n_k>λ>1$, we define a new sequence $\{m_k\}$ formed by all possible $l$-wise sums $\pm n_{k_1}\pm n_{k_2}\pm \ldots\pm n_{k_l}$. We prove if $λ>λ_l$, then any series \begin{equation} \sum_kc_ke^{im_kx},\qquad (1) \end{equation} with $\sum_k|c_k|^2<\infty$ converges almost everywhere after any rearrangement of the terms, where $1<λ_l<2$ is a certain critical value. We establish this property, proving a new Khintchine type inequality $\|S\|_p\le C_{l,λ,p}\|S\|_2$, $p>2$, where $S$ is a finite sum of form (1). For $λ\ge 3$, we also establish a sharp rate $p^{l/2}$ for the growth of the constant $C_{l,λ,p}$ as $p\to\infty$. Such an estimate for the Rademacher chaos sums was proved independently by Bonami and Kiener. In the case of $λ\ge 3$ we also establish some inverse convergence properties of series (1): 1) if series (1) converges a.e., then $\sum_k|c_k|^2<\infty$, 2) if it a.e. converges to zero, then $c_k=0$.
