On some sampling-related frames in U-invariant spaces

This paper concerns the characterization as frames of some sequences in U-invariant spaces of a separable Hilbert space H where U denotes an unitary operator defined on 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 L2 (R), where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In so doing, we need that the unitary operator U belongs to a continuous group of unitary operators.

Generalized sampling in U-invariant subspaces

In this work we carry out some results in sampling theory for U-invariant subspaces of a separable Hilbert space H, also called atomic subspaces. These spaces are a generalization of the well-known shift- invariant subspaces in L2 (R); here the space L2 (R) is replaced by H, and the shift operator by U. Having as data the samples of some related operators, we derive frame expansions allowing the recovery of the elements in Aa. Moreover, we include a frame perturbation-type result whenever the samples are affected with a jitter error.