Geodesic
Bessel Sequences as Projections of Orthogonal Systems
Matematičeskie zametki, Tome 81 (2007) no. 6, pp. 893-903 Cet article a éte moissonné depuis la source Math-Net.Ru

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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
Mots-clés : Gram matrix. 
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     author = {S. Ya. Novikov},
     title = {Bessel {Sequences} as {Projections} of {Orthogonal} {Systems}},
     journal = {Matemati\v{c}eskie zametki},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2007_81_6_a7/}
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S. Ya. Novikov. Bessel Sequences as Projections of Orthogonal Systems. Matematičeskie zametki, Tome 81 (2007) no. 6, pp. 893-903. http://geodesic.mathdoc.fr/item/MZM_2007_81_6_a7/

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