Generalized Fourier series and shift-invariant subspaces

A principal shift invariant subspace of $L^{2}(\IR)$ is isometric to a weighted norm space $L^{2}(\IT, w)$. Using results obtained earlier by the author on the basis properties of subsystems of the trigonometric system in the weighted norm spaces $L^{2}(\IT, w)$ we obtain corresponding results concerning generators of principal shift invariant subspaces.