Using a natural deconvolution for analysis of perturbed integer sampling in shift-invariant spaces

An important cornerstone of both wavelet and sampling theory is shift-invariant spaces, that is, spaces V spanned by a Riesz basis of integer-translates of a single function. Under some mild differentiability and decay assumptions on the Fourier transform of this function, we show that V also is generated by a function with Fourier transform φ ˆ ( ξ ) = ∫ ξ − π ξ + π g ( ν ) d ν for some g with ∫ R g ( ξ ) d ξ = 1 . We explain why analysis of this particular generating function can be more likely to provide large jitter bounds ε such that any f ∈ V can be reconstructed from perturbed integer samples f ( k + ε k ) whenever sup k ∈ Z | ε k | ⩽ ε . We use this natural deconvolution of φ ˆ ( ξ ) to further develop analysis techniques from a previous paper. Then we demonstrate the resulting analysis method on the class of spaces for which g has compact support and bounded variation (including all spaces generated by Meyer wavelet scaling functions), on some particular choices of φ for which we know of no previously published bounds and finally, we use it to improve some previously known bounds for B-spline shift-invariant spaces.