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Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 1, Pages 33–58 (Mi izv417)  

This article is cited in 6 scientific papers (total in 6 papers)

A dyadic analogue of Wiener's Tauberian theorem and some related questions

B. I. Golubov

Moscow Engineering Physics Institute (State University)

Abstract: A dyadic analogue is proved of Wiener's Tauberian convolution theorem for two functions. Closedness criteria are established for the linear span of the set of binary shifts $\{f( \circ\oplus y)\colon y\geqslant 0\}$ for a given function $f\in L(\mathbb R_+)$ or $f\in L^2(\mathbb R_+)$. A consequence of these criteria is that the linear span of the set of binary shifts $\{f( \circ\oplus y)\colon 0\leqslant y\leqslant 1\}$ for a given function $f\in L([0,1))$ ($f\in L^2([0,1))$) is dense in the space $L([0,1))$ ($L^2([0,1))$) if and only if all the Fourier coefficients of $f$ with respect to the orthonormalized Walsh system on $[0,1)$ are non-zero.

DOI: https://doi.org/10.4213/im417

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English version:
Izvestiya: Mathematics, 2003, 67:1, 29–53

Bibliographic databases:

UDC: 517.5
MSC: 11M45, 30B50, 40E05, 42A32, 42A38, 42C10, 42C10, 44A10, 47A10
Received: 15.03.2002

Citation: B. I. Golubov, “A dyadic analogue of Wiener's Tauberian theorem and some related questions”, Izv. RAN. Ser. Mat., 67:1 (2003), 33–58; Izv. Math., 67:1 (2003), 29–53

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. B. I. Golubov, “Fractional Modified Dyadic Integral and Derivative on $\mathbb{R}_+$”, Funct. Anal. Appl., 39:2 (2005), 64–70  mathnet  crossref  crossref  mathscinet  zmath
    2. B. I. Golubov, “Modified Dyadic Integral and Fractional Derivative on $\mathbb R_+$”, Math. Notes, 79:2 (2006), 196–214  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Golubov B.I., “On approximation by convolutions and bases of shifts of a function”, Anal. Math., 34:1 (2008), 9–28  crossref  mathscinet  zmath  isi  elib  scopus
    4. S. S. Volosivets, “Applications of $\mathbf P$-adic generalized functions and approximations by a system of $\mathbf P$-adic translations of a function”, Siberian Math. J., 50:1 (2009), 1–13  mathnet  crossref  mathscinet  isi  elib
    5. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196  mathnet  crossref  crossref  isi  elib  elib
    6. S. S. Platonov, “An Analog of Titchmarsh's Theorem for the Fourier–Walsh Transform”, Math. Notes, 103:1 (2018), 96–103  mathnet  crossref  crossref  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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