ON A PROPERTY OF GENERAL HAAR SYSTEM

Abstract

In the paper we prove that for some type of general Haar systems (particularly for classical Haar system) and for any $\varepsilon>0$ there exists a set {$E\subset (0,1)^2,|E|>1-\varepsilon$}, such that for every $f\in L^1(0,1)^2$ one can find a function $g\in L^1(0,1)^2$, which coincides with $f$ on $E$ and Fourier--Haar coefficients $\{c_{(i,k)}(g)\}_{i,k=1}^\infty$ are monotonic over all rays.