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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 4, Pages 157–188 (Mi izv4055)  

This article is cited in 2 scientific papers (total in 2 papers)

Quasi-measures, Hausdorff $p$-measures and Walsh and Haar series

M. G. Plotnikov

Vologda State Academy of Milk Industry

Abstract: 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: dyadic group, Haar series, Walsh series, uniqueness set.

DOI: https://doi.org/10.4213/im4055

Full text: PDF file (793 kB)
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English version:
Izvestiya: Mathematics, 2010, 74:4, 819–848

Bibliographic databases:

UDC: 517.518.3
MSC: 42C25, 42C10, 42B05
Received: 21.10.2008

Citation: M. G. Plotnikov, “Quasi-measures, Hausdorff $p$-measures and Walsh and Haar series”, Izv. RAN. Ser. Mat., 74:4 (2010), 157–188; Izv. Math., 74:4 (2010), 819–848

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. G. Plotnikov, Yu. A. Plotnikova, “Decomposition of dyadic measures and unions of closed $\mathscr{U}$-sets for series in a Haar system”, Sb. Math., 207:3 (2016), 444–457  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. V. A. Skvortsov, “Integration of Banach-valued functions and Haar series with Banach-valued coefficients”, Moscow University Mathematics Bulletin, 72:1 (2017), 24–30  mathnet  crossref  mathscinet  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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