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Ufimsk. Mat. Zh., 2011, Volume 3, Issue 1, Pages 47–52 (Mi ufa81)  

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

Riesz bases in weighted spaces

A. A. Putintseva

Bashkir State University, Ufa, Russia

Abstract: The article deals with weighted Hilbert spaces with convex weights. Let $h$ be a convex function on a bounded interval $I$ of the real axis. We denote a space of locally integrable functions on $I$, such that
$$ \|f\|:=\sqrt{\int _I|f(t)|^2e^{-2h(t)} dt}<\infty $$
by $L_2(I,h)$.
If $I=(-\pi;\pi)$, $h(t)\equiv1$, the space $L_2(I,h)$ coincides with the classical space $L_2(-\pi;\pi)$ and the Fourier trigonometric system is a Riesz basis in this space. As it has been shown by B. J. Levin, nonharmonic Riesz bases in $L_2(-\pi;\pi)$ can be constructed using a system of zeros of entire functions of sine type. In this paper we prove that if a Riesz basis of exponentials exists in the space $L_2(I,h)$, this space is isomorphic (as a normed space) to the classical space $L_2(I)$. Thus, the existence of Riesz bases of exponentials is the exclusive property of the classical space $L_2(-\pi;\pi)$.

Keywords: Riesz basis, weighted Hilbert spaces, reproducing kernel, Fourier–Laplace transform, functions оf sine type.

Full text: PDF file (370 kB)
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English version:
Ufa Mathematical Journal, 2011, 3:1, 45–50 (PDF, 374 kB)

Bibliographic databases:

Document Type: Article
UDC: 517.5
Received: 03.02.2011

Citation: A. A. Putintseva, “Riesz bases in weighted spaces”, Ufimsk. Mat. Zh., 3:1 (2011), 47–52; Ufa Math. J., 3:1 (2011), 45–50

Citation in format AMSBIB
\Bibitem{Put11}
\by A.~A.~Putintseva
\paper Riesz bases in weighted spaces
\jour Ufimsk. Mat. Zh.
\yr 2011
\vol 3
\issue 1
\pages 47--52
\mathnet{http://mi.mathnet.ru/ufa81}
\zmath{https://zbmath.org/?q=an:1240.46022}
\transl
\jour Ufa Math. J.
\yr 2011
\vol 3
\issue 1
\pages 45--50


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    This publication is cited in the following articles:
    1. A. R. Bagautdinova, A. V. Lutsenko, V. I. Lutsenko, E. D. Shaimuratova, “Integralnye otsenki proizvodnykh analiticheskikh funktsii vne vypuklykh oblastei”, Ufimsk. matem. zhurn., 4:4 (2012), 13–21  mathnet  mathscinet
    2. K. P. Isaev, A. V. Lutsenko, R. S. Yulmukhametov, “On unconditional exponential bases in weak weighted spaces on segment”, Ufa Math. J., 8:4 (2016), 88–97  mathnet  crossref  isi  elib
    3. K. P. Isaev, R. S. Yulmukhametov, A. A. Yunusov, “On unconditional exponential bases in weighted spaces on interval of real axis”, St. Petersburg Math. J., 28:5 (2017), 689–706  mathnet  crossref  mathscinet  isi  elib
    4. Isaev K.P., “On Unconditional Exponential Bases in Weighted Spaces on Interval of Real Axis”, Lobachevskii J. Math., 38:1 (2017), 48–61  crossref  mathscinet  zmath  isi  scopus
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