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Izv. Akad. Nauk SSSR Ser. Mat., 1985, Volume 49, Issue 2, Pages 243–282 (Mi izv1354)  

This article is cited in 11 scientific papers (total in 11 papers)

Two criteria for weak generalized localization for multiple trigonometric Fourier series of functions in $L_p$, $p\geqslant1$

I. L. Bloshanskii

Abstract: The concept of weak generalized localization almost everywhere is introduced. For the multiple Fourier series of a function $f$, weak generalized localization almost everywhere holds on the set $E$ ($E$ is an arbitrary set of positive measure $E\subset T^N=[-\pi,\pi]^N$) if the condition $f(x)\in L_p(T^N)$, $p\geqslant1$, $f=0$ on $E$ implies that the indicated series converges almost everywhere on some subset $E_1\subset E$ of positive measure. For a large class of sets $\{E\}$, $E\subset T^N$, a number of propositions are proved showing that weak localization of rectangular sums holds on the set $E$ in the classes $L_p$, $p\geqslant1$, if and only if the set $E$ has certain specific properties. In the course of the proof the precise geometry and structure of the subset $E_1$ of $E$ on which the multiple Fourier series converges almost everywhere to zero are determined.
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1986, 26:2, 223–262

Bibliographic databases:

UDC: 517.5
MSC: 42B05
Received: 25.04.1983

Citation: I. L. Bloshanskii, “Two criteria for weak generalized localization for multiple trigonometric Fourier series of functions in $L_p$, $p\geqslant1$”, Izv. Akad. Nauk SSSR Ser. Mat., 49:2 (1985), 243–282; Math. USSR-Izv., 26:2 (1986), 223–262

Citation in format AMSBIB
\by I.~L.~Bloshanskii
\paper Two criteria for weak generalized localization for multiple trigonometric Fourier series of functions in $L_p$, $p\geqslant1$
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1985
\vol 49
\issue 2
\pages 243--282
\jour Math. USSR-Izv.
\yr 1986
\vol 26
\issue 2
\pages 223--262

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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. I. L. Bloshanskii, “The structure and geometry of maximal sets of convergence and unbounded divergence almost everywhere of multiple Fourier series of functions in $L_1$ equal to zero on a given set”, Math. USSR-Izv., 35:1 (1990), 1–35  mathnet  crossref  mathscinet  zmath
    2. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. S. K. Bloshanskaya, I. L. Bloshanskii, “Generalized localization for the multiple Walsh–Fourier series of functions in $L_p$, $p\geqslant 1$”, Sb. Math., 186:2 (1995), 181–196  mathnet  crossref  mathscinet  zmath  isi
    4. S. K. Bloshanskaya, I. L. Bloshanskii, T. Yu. Roslova, “Generalized localization for the double trigonometric Fourier series and the Walsh–Fourier series of functions in $L\log^+L\log^+\log^+L$”, Sb. Math., 189:5 (1998), 657–682  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. O. K. Ivanova, “Majorant estimates for partial sums of multiple Fourier series from Orlicz spaces that vanish on some set”, Math. Notes, 65:6 (1999), 694–700  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. I. L. Bloshanskii, “A Criterion for Weak Generalized Localization in the Class $L_1$ for Multiple Trigonometric Series from the Viewpoint of Isometric Transformations”, Math. Notes, 71:4 (2002), 464–476  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. I. L. Bloshanskii, T. A. Matseevich, “A Weak Generalize Localization of Multiple Fourier Series of Continuous Functions with a Certain Module of Continuity”, Journal of Mathematical Sciences, 155:1 (2008), 31–46  mathnet  crossref  mathscinet  zmath
    8. I. L. Bloshanskii, O. V. Lifantseva, “Weak Generalized Localization for Multiple Fourier Series Whose Rectangular Partial Sums Are Considered with Respect to Some Subsequence”, Math. Notes, 84:3 (2008), 314–327  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. O. V. Lifantseva, “Necessary Conditions for the Weak Generalized Localization of Fourier Series with “Lacunary Sequence of Partial Sums””, Math. Notes, 86:3 (2009), 373–384  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. S. K. Bloshanskaya, I. L. Bloshanskii, O. V. Lifantseva, “Trigonometric Fourier Series and Walsh–Fourier Series with Lacunary Sequence of Partial Sums”, Math. Notes, 93:2 (2013), 332–336  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. S. K. Bloshanskaya, I. L. Bloshanskii, “A weak generalized localization criterion for multiple Walsh–Fourier series with $J_k$-lacunary sequence of rectangular partial sums”, Proc. Steklov Inst. Math., 285 (2014), 34–55  mathnet  crossref  crossref  isi  elib  elib
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