On Some Sampling-Related Frames in U -Invariant Spaces

Abstract

This paper is concerned with the characterization as frames of some sequences in $U$-invariant spaces of a separable Hilbert space $\mathscr{H}$ where $U$ denotes an unitary operator defined on $\mathscr{H}$; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces in $L^2\left(ℝ\right)$, where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operator $U$ to belong to a continuous group of unitary operators.