On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian

Abstract

Let λ be a positive number, and let (x j : j ∈ Z) ⊂ R be a fixed Riesz-basis sequence, namely, (x j ) is strictly increasing, and the set of functions {R t → e ix j t

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

“…Later, Schlumprecht and Sivakumar (see [2]), were able to prove a similar result by using parametrized scattered translates of the Gaussian rather than tempered splines. A natural question to ask is that of whether one may use other families of interpolants to produce similar recovery results.…”

“…We may now take a similar path as the one laid out in [1], as well as [2]. We introduce the notation:…”

Kernel Approximation on Manifolds II: The $L_{\infty}$ Norm of the $L_2$ Projector

“…Remark 4.2. In [6,17] it is shown that invertible matrices that decay exponentially away from the diagonal have inverses with the same property. However, it is not clear that their results are applicable to this situation.…”

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 .…”

“…The last inequality comes from summing the geometric series and applying (8). This is the desired result once we take square roots.…”