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

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

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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• http://mi.mathnet.ru/eng/izv417
• https://doi.org/10.4213/im417
• http://mi.mathnet.ru/eng/izv/v67/i1/p33

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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
2. B. I. Golubov, “Modified Dyadic Integral and Fractional Derivative on $\mathbb R_+$”, Math. Notes, 79:2 (2006), 196–214
3. Golubov B.I., “On approximation by convolutions and bases of shifts of a function”, Anal. Math., 34:1 (2008), 9–28
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
5. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196
6. S. S. Platonov, “An Analog of Titchmarsh's Theorem for the Fourier–Walsh Transform”, Math. Notes, 103:1 (2018), 96–103
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