On Haar series of A–integrable functions

Abstract

In this paper we obtain a necessary and sufficient condition on the sequence of natural numbers {q n } such that the almost everywhere convergence of the cubic partial sums S qn (x) of the multiple Haar series Σn a nχn(x) and the condition lim inf $$\lambda \cdot mes\left\{ {x:\begin{array}{*{20}{c}} {\sup } \\ n \end{array}\left| {S{}_{qn}\left( x \right)} \right| \succ \lambda } \right\} = 0$$ , imply that the coefficients a n can be uniquely determined by the sum of the series. Also, we have obtained a necessary and sufficient condition for the series $$\sum\limits_{n = 1}^\infty {{\varepsilon _n}{a_n}} {\chi _n}\left( x \right)$$ with an arbitrary bounded sequence {e n} to be a Fourier-Haar series of an A-integrable function.