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 L2(I) are projections of orthogonal systems from the wider space L2(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 L2(J) onto L2)(I) is a tight frame, but not a basis.
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