Bessel Sequences as Projections of Orthogonal Systems

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.