Geodesic
Quasi-measures, Hausdorff $p$-measures and Walsh and Haar series
Izvestiya. Mathematics , Tome 74 (2010) no. 4, pp. 819-848

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We study the classes of multiple Haar and Walsh series with at most polynomial growth of the rectangular partial sums. In terms of the Hausdorff $p$-measure, we find a sufficient condition (a criterion for the multiple Haar series) for a given set to be a $U$-set for series in the given class. We solve the recovery problem for the coefficients of the series in this class converging outside a uniqueness set. A Bari-type theorem is proved for the relative uniqueness sets for multiple Haar series. For one-dimensional Haar series, we get a criterion for a given set to be a $U$-set under certain assumptions that generalize the Arutyunyan–Talalyan conditions. We study the problem of describing those Cantor-type sets that are relative uniqueness sets for Haar series.
Keywords: Haar series, Walsh series, uniqueness set.
Mots-clés : dyadic group 
@article{IM2_2010_74_4_a8,
     author = {M. G. Plotnikov},
     title = {Quasi-measures, {Hausdorff} $p$-measures and {Walsh} and {Haar} series},
     journal = {Izvestiya. Mathematics },
     pages = {819--848},
     publisher = {mathdoc},
     volume = {74},
     number = {4},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2010_74_4_a8/}
}
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M. G. Plotnikov. Quasi-measures, Hausdorff $p$-measures and Walsh and Haar series. Izvestiya. Mathematics , Tome 74 (2010) no. 4, pp. 819-848. http://geodesic.mathdoc.fr/item/IM2_2010_74_4_a8/