Uniqueness for multiple Haar series

Uniqueness questions are considered for multiple Haar series convergent over rectangles or in the sense of $\rho$-regular convergence. A condition is found ensuring that a given set is a relative uniqueness set under assumptions that are many-dimensional analogues of the Arutyunyan–Talalyan condition. This generalizes to $\rho$-regular convergence results for convergence over rectangles obtained by Movsisyan and Skvortsov. A monotonicity theorem is proved under very general assumptions for a dyadic-interval function used in the construction of a many-dimensional generalized integral of Perron type, which is called the $(P^{\rho,*}_d )$-integral. With the help of this integral one can recover by Fourier's formulae the coefficients of multiple Haar series from a fairly broad class including, in particular, series with power growth of partial sums at points with at least one dyadic rational coordinate. It is observed that already in the two-dimensional case the main results are best possible in a certain sense.