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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2007, Volume 19, Issue 3, Pages 1–75 (Mi aa119)

Research Papers

Spectral subspaces of $L^p$ for $p<1$

A. B. Aleksandrov

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: Let $\Omega$ be an open subset of $\mathbb{R}^n$. Denote by $L^p_{\Omega}(\mathbb{R}^n)$ the closure in $L^p(\mathbb{R}^n)$ of the set of all functions $\varepsilon\in L^1(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ whose Fourier transform has compact support contained in $\Omega$. The subspaces of the form $L^p_\Omega(\mathbb{R}^n)$ are called the spectral subspaces of $L^p(\mathbb{R}^n)$. It is easily seen that each spectral subspace is translation invariant; i.e., $f(x+a)\in L^p_\Omega(\mathbb{R}^n)$ for all $f\in L^p_\Omega(\mathbb{R}^n)$ and $a\in\mathbb{R}^n$. Sufficient conditions are given for the coincidence of $L^p_\Omega(\mathbb{R}^n)$ and $L^p(\mathbb{R}^n)$. In particular, an example of a set $\Omega$ is constructed such that the above spaces coincide for sufficiently small $p$ but not for all $p\in(0,1)$. Moreover, the boundedness of the functional $f\mapsto(\mathcal{F} f)(a)$ with $a\in\Omega$, which is defined initially for sufficiently “good” functions in $L^p_\Omega(\mathbb{R}^n)$, is investigated. In particular, estimates of the norm of this functional are obtained. Also, similar questions are considered for spectral subspaces of $L^p(G)$, where $G$ is a locally compact Abelian group.

Keywords: Translation invariant subspace, spectral subspace, Hardy classes, uniqueness set.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:3, 327–374

Bibliographic databases:

MSC: 42B35

Citation: A. B. Aleksandrov, “Spectral subspaces of $L^p$ for $p<1$”, Algebra i Analiz, 19:3 (2007), 1–75; St. Petersburg Math. J., 19:3 (2008), 327–374

Citation in format AMSBIB
\Bibitem{Ale07}
\by A.~B.~Aleksandrov
\paper Spectral subspaces of~$L^p$ for $p<1$
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 3
\pages 1--75
\mathnet{http://mi.mathnet.ru/aa119}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2340705}
\zmath{https://zbmath.org/?q=an:1202.42045}
\transl
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 3
\pages 327--374
\crossref{https://doi.org/10.1090/S1061-0022-08-01001-7}

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This publication is cited in the following articles:
1. A. B. Aleksandrov, “Approximation in $L^p(\mathbb R^d)$, $0<p<1$, by linear combinations of the characteristic functions of balls”, J. Math. Sci. (N. Y.), 165:4 (2010), 431–434
2. P. Ivanishvili, S. V. Kislyakov, “Correction up to a function with sparse spectrum and uniformly convergent Fourier series”, J. Math. Sci. (N. Y.), 172:2 (2011), 195–206
3. S. V. Kislyakov, “Ispravlenie do funktsii s redkim spektrom i ravnomerno skhodyaschimsya integralom Fure v sluchae gruppy $\mathbb R^n$”, Issledovaniya po lineinym operatoram i teorii funktsii. 46, Zap. nauchn. sem. POMI, 467, POMI, SPb., 2018, 116–127
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