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Mat. Sb., 2005, Volume 196, Number 2, Pages 97–116 (Mi msb1268)  

This article is cited in 4 scientific papers (total in 4 papers)

Uniqueness for multiple Haar series

M. G. Plotnikov

Vologda State Academy of Milk Industry

Abstract: 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.

DOI: https://doi.org/10.4213/sm1268

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English version:
Sbornik: Mathematics, 2005, 196:2, 243–261

Bibliographic databases:

UDC: 517.518.3
MSC: 42B05, 42C10, 40A05
Received: 04.11.2003 and 30.08.2004

Citation: M. G. Plotnikov, “Uniqueness for multiple Haar series”, Mat. Sb., 196:2 (2005), 97–116; Sb. Math., 196:2 (2005), 243–261

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, “Several properties of generalized multivariate integrals and theorems of the du Bois-Reymond type for Haar series”, Sb. Math., 198:7 (2007), 967–991  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. M. G. Plotnikov, “Quasi-measures, Hausdorff $p$-measures and Walsh and Haar series”, Izv. Math., 74:4 (2010), 819–848  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. M. G. Plotnikov, Yu. A. Plotnikova, “Martingaly i teoremy Kantora–Yunga–Bernshteina i Valle-Pussena”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:4(2) (2014), 569–574  mathnet  crossref  elib
    4. Plotnikov M., Plotnikova J., “Non-uniqueness for rearranged double haar series”, Anal. Math., 42:2 (2016), 173–184  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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