Recovery of Paley–Wiener functions using scattered translates of regular interpolators

On the structure and interpolation properties of quasi shift-invariant spaces

“…Consequently, Theorem 5 recovers Theorem 1 of [45], which constitutes the special case when ψ is the sinc function, whose Fourier transform is the characteristic function of T. Also regard that the proof follows from Proposition 3(i) in this case given the assumptions on ψ. Examples of regular interpolators may be found in Section 5 of [45] and Section 8 of [28], but here we note that a prominent example is the family of Gaussian generators: φ α (x) = e −|x/α| 2 , α ≥ 1.…”

“…For additional convergence phenomena similar to the ones listed in this section, the interested reader is referred to [28,29,32,45,47,48,51,61].…”

“…These works, and the results of [45], which solve Problem 2 in the special case when ψ = sinc and the exponentials (e −ixj · ) j∈Z are a Riesz basis for L 2 (T), form the inspiration for our study of this problem. More precisely, Theorems 1 and 2 in [45] give conditions on families of generators (φ α ) which allow for recovery of f ∈ P W π = V (sinc, Z) via their interpolants in V (φ α , Y).…”

Recovery of bivariate band-limited functions using scattered translates of the Poisson kernel

“…Unfortunately, we do not obtain results as strong as the ones found in [6,8,5], which allow us to recover f ∈ PW [−π,π] in L 2 and uniformly; however, if F[ f ] is concentrated enough near the origin, a recovery result similar to [2] is possible. With this in mind, for 0 < β < δ, we let B β = {ξ ∈ R 2 : |ξ | ≤ β} be the ball of radius β in R 2 .…”

“…In fact, Lyubarskii and Madych showed in [6] that one may use tempered splines to solve this problem, while Schlumprecht and Sivakumar showed in [8] that the same is true using scattered translates of the Gaussian. Along the same lines, the author showed in [5] that one may use scattered translates of 'regular interpolators' to solve this problem. In particular, one may use the Poisson kernel.…”