  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
$d=1,2,\dots$.
