Geodesic
Uniqueness for multiple Haar series
Sbornik. Mathematics, Tome 196 (2005) no. 2, pp. 243-261 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {M. G. Plotnikov},
     title = {Uniqueness for multiple {Haar} series},
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     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_2_a4/}
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M. G. Plotnikov. Uniqueness for multiple Haar series. Sbornik. Mathematics, Tome 196 (2005) no. 2, pp. 243-261. http://geodesic.mathdoc.fr/item/SM_2005_196_2_a4/

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