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Mat. Zametki, 2007, Volume 81, Issue 6, Pages 893–903 (Mi mz3739)  

This article is cited in 7 scientific papers (total in 7 papers)

Bessel Sequences as Projections of Orthogonal Systems

S. Ya. Novikov

Samara State University

Abstract: We prove generalizations of the Schur and Olevskii theorems on the continuation of systems of functions from an interval $I$ to orthogonal systems on an interval $J$, $I\subset J$. Only Bessel systems in $L^2(I)$ are projections of orthogonal systems from the wider space $L^2(J)$. This fact allows us to use a certain method for transferring the classical theorems on the almost everywhere convergence of orthogonal series (the Menshov–Rademacher, Paley–Zygmund, and Garcia theorems) to series in Bessel systems. The projection of a complete orthogonal system from $L^2(J)$ onto $L^2(I)$ is a tight frame, but not a basis.

Keywords: Bessel sequence, orthogonal system, tight frame, complex Hilbert space, Schur criterion, Menshov–Rademacher theorem, Paley–Zygmund theorem, Gram matrix

DOI: https://doi.org/10.4213/mzm3739

Full text: PDF file (511 kB)
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English version:
Mathematical Notes, 2007, 81:6, 800–809

Bibliographic databases:

UDC: 517.51+517.98
Received: 20.03.2006
Revised: 26.09.2006

Citation: S. Ya. Novikov, “Bessel Sequences as Projections of Orthogonal Systems”, Mat. Zametki, 81:6 (2007), 893–903; Math. Notes, 81:6 (2007), 800–809

Citation in format AMSBIB
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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. P. A. Terekhin, “Proektsionnye kharakteristiki besselevykh sistem”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 9:1 (2009), 44–51  mathnet  elib
    2. P. A. Terekhin, “Linear algorithms of affine synthesis in the Lebesgue space $L^1[0,1]$”, Izv. Math., 74:5 (2010), 993–1022  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. M. I. Ismailov, “Gilbertovy obobscheniya $b$-besselevykh sistem”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 11:3(2) (2011), 3–10  mathnet  elib
    4. P. A. Terekhin, “On Bessel Systems in a Banach Space”, Math. Notes, 91:2 (2012), 272–282  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. P. A. Terekhin, “Affinnye sistemy funktsii tipa Uolsha. Ortogonalizatsiya i popolnenie”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:4(1) (2014), 395–400  mathnet
    6. Ismailov M.I., Nasibov Y.I., “on One Generalization of Banach Frame”, Azerbaijan J. Math., 6:2 (2016), 143–159  mathscinet  zmath  isi  elib
    7. Ismailov M. Guliyeva F. Nasibov Yu., “On a Generalization of the Hilbert Frame Generated by the Bilinear Mapping”, J. Funct. space, 2016, 9516839  crossref  mathscinet  zmath  isi  elib  scopus
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