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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
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
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
Linking options:
https://www.mathnet.ru/eng/mzm3739https://doi.org/10.4213/mzm3739 https://www.mathnet.ru/eng/mzm/v81/i6/p893
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Abstract page: | 485 | Full-text PDF : | 204 | References: | 67 | First page: | 11 |
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