论文标题
$ l^p $空间等效于HAAR系统的扩张和翻译序列
Sequences of dilations and translations equivalent to the Haar system in $L^p$-spaces
论文作者
论文摘要
令$ f = \ sum_ {k = 0}^{\ infty} c_kh_ {2^k} $，其中$ \ {h_n \} $是经典的haar系统，$ c_k \ in \ mathbb {c} $。给定（1，\ infty）$中的$ p \，我们找到了鲜明的条件，在此条件下，序列$ \ {f_n \} _ {n = 1}^\ infty $的扩张和$ f $的翻译是$ l^p [0,1] $的基础，等于$ l^p [0,1] $，等于$ \ \ \ {h_n \} n = 1}获得的结果很大程度上取决于$ p \ ge 2 $还是$ 1 <p <2 $，并包括$ l_p $ - scale的端点$ bmo_d $和$ h_d^1 $。这些证明是基于适当的分配积极整数$ \ mathbb {n} = \ cup_ {d = 1}^\ infty n_d $，以使$ \ {f_n \} _ {f_n \} _ {n = 1}^\ infty n = n = 1}^\ infty y in n a $ l_p $的事实$ \ {f_n \} _ {n_d} $是子空间$ [h_ {m}，m \ in n_d] _ {l_p} $的基础，等于haar subsequence $ \ \ \ {h_n \} _ {
Let $f=\sum_{k=0}^{\infty}c_kh_{2^k}$, where $\{h_n\}$ is the classical Haar system, $c_k\in\mathbb{C}$. Given a $p\in (1,\infty)$, we find the sharp conditions, under which the sequence $\{f_n\}_{n=1}^\infty$ of dilations and translations of $f$ is a basis in the space $L^p[0,1]$, equivalent to $\{h_n\}_{n=1}^\infty$. The results obtained depend substantially on whether $p\ge 2$ or $1<p<2$ and include as the endpoints of the $L_p$-scale the spaces $BMO_d$ and $H_d^1$. The proofs are based on an appropriate splitting the set of positive integers $\mathbb{N}=\cup_{d=1}^\infty N_d$ so that the equivalence of $\{f_n\}_{n=1}^\infty$ to the Haar system in $L_p$ would be ensured by the fact that $\{f_n\}_{n\in N_d}$ is a basis in the subspace $[h_{m},m\in N_d]_{L_p}$, equivalent to the Haar subsequence $\{h_n\}_{n\in N_d}$ for every $d=1,2,\dots$.
