Sequences of dilations and translations equivalent to the Haar system in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e25" altimg="si25.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math>-spaces

Let f=∑k=0∞ckh2k, where {hn} is the classical Haar system, ck∈ℂ. Given a p∈(1,∞), we find sharp conditions, under which sequence {fn}n=1∞ of dilations and translations f basis in space Lp[0,1], equivalent to {hn}n=1∞. The results obtained depend substantially on whether p≥2 or 1<p<2 include as endpoints Lp-scale spaces BMOd Hd1. proofs are based an appropriate splitting set positive integers N=∪d=1∞Nd so that equivalence system Lp would be ensured by fact {fn}n∈Nd subspace [hm,m∈Nd]Lp, subsequence {hn}n∈Nd for every d=1,2,….