Generalized Irregular Sampling in Shift-Invariant Spaces

This article concerns the problem of stable recovering of any function in a shiftinvariant space from irregular samples of some filtered versions of the function itself. These samples arise as a perturbation of regular samples. The starting point is the generalized regular sampling theory which allows to recover any function f in a shiftinvariant space from the samples at {rn}n∈Z of s filtered versions L1f,L2f, . . . ,Lsf of f , where the number of channels s is greater or equal than the sampling period r. These regular samples can be expressed as the frame coefficients of a related to f function in L(0, 1) with respect to certain frame for L(0, 1). The irregular samples are also obtained as a perturbation of the aforesaid frame. As a natural consequence, the irregular sampling results arise from the theory of perturbation of frames. The paper ends putting the theory to work in some spline examples where Kadec-type results are obtained.