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Bessel sequences as projections of orthogonal systems

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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 ⊃ 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 Men’shov-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.

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Authors and Affiliations

  1. Samara State University, Russia

    S. Ya. Novikov

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  1. S. Ya. Novikov
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Original Russian Text © S. Ya. Novikov, 2007, published in Matematicheskie Zametki, 2007, Vol. 81, No. 6, pp. 893–903.

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Novikov, S.Y. Bessel sequences as projections of orthogonal systems. Math Notes 81, 800–809 (2007). https://doi.org/10.1134/S0001434607050276

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  • DOI: https://doi.org/10.1134/S0001434607050276

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