Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces

Under the appropriate definition of sampling density D_ϕ, a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if D_ϕ≥1. This result is similar to Landau's result for the Paley-Wiener space B _1/2 . If the shift invariant space consists of polynomial splines, then we show that D_ϕ<1 is sufficient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B _1/2 .