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Matematicheskie Zametki, 2007, Volume 81, Issue 6, Pages 893–903
DOI: https://doi.org/10.4213/mzm3739
(Mi mzm3739)
 

This article is cited in 10 scientific papers (total in 10 papers)

Bessel Sequences as Projections of Orthogonal Systems

S. Ya. Novikov

Samara State University
References:
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.
Received: 20.03.2006
Revised: 26.09.2006
English version:
Mathematical Notes, 2007, Volume 81, Issue 6, Pages 800–809
DOI: https://doi.org/10.1134/S0001434607050276
Bibliographic databases:
UDC: 517.51+517.98
Language: Russian
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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Linking options:
  • https://www.mathnet.ru/eng/mzm3739
  • https://doi.org/10.4213/mzm3739
  • https://www.mathnet.ru/eng/mzm/v81/i6/p893
  • This publication is cited in the following 10 articles:
    1. S. Ya. Novikov, V. V. Sevost'yanova, “Maltsev equal-norm tight frames”, Izv. Math., 86:4 (2022), 770–781  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. S Ya Novikov, M E Fedina, “Naimark complements in the building of simplices”, J. Phys.: Conf. Ser., 1745:1 (2021), 012108  crossref
    3. Novikov S.Ya., “Processing of Sparse Signals and Mutual Coherence of “Measurable” Vectors”, Lobachevskii J. Math., 41:4, SI (2020), 666–675  crossref  mathscinet  isi
    4. Ismailov M.I., Nasibov Y.I., “on One Generalization of Banach Frame”, Azerbaijan J. Math., 6:2 (2016), 143–159  mathscinet  zmath  isi  elib
    5. 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
    6. 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  crossref  elib
    7. P. A. Terekhin, “On Bessel Systems in a Banach Space”, Math. Notes, 91:2 (2012), 272–282  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. 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  crossref  elib
    9. 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
    10. P. A. Terekhin, “Proektsionnye kharakteristiki besselevykh sistem”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 9:1 (2009), 44–51  mathnet  crossref  elib
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