Sequences of dilations and translations equivalent to the Haar system in $L^p$-spaces

Abstract

Let f = ∞ k=0 c k h 2 k , where {h n } is the classical Haar system, c k ∈ C. Given a p ∈ (1, ∞), we find the sharp conditions, under which the sequence {f n } ∞ n=1 of dilations and translations of f is a basis in the space L p [0, 1], equivalent to {h n } ∞ n=1 . The results obtained depend substantially on whether p ≥ 2 or 1 < p < 2 and include as the endpoints of the L p -scale the spaces BM O d and H 1 d . The proofs are based on an appropriate splitting the set of positive integers N = ∪ ∞ d=1 N d so that the equivalence of {f n } ∞ n=1 to the Haar system in L p would be ensured by the fact that {f n } n∈N d is a basis in the subspace [h m , m ∈ N d ] Lp , equivalent to the Haar subsequence {h n } n∈N d for every d = 1, 2, .