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 ϕ with Fourier transform equal to the convolution of g with the characteristic function living on the interval [-pi,pi]. 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 the supremum of |ε_k| is smaller than ε. We use this natural deconvolution 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.