The recovery of irregularly sampled band limited functions via tempered splines (Q1340827)

The paper is devoted to the cardinal spline interpolation problem \(s(x_n) = f(x_n)\), \(n\in \mathbb{Z}\), by tempered splines \(s(x)\) of order \(2k\) and sequence of nodes \(\{x_n\}\) such that the functions \(\{e^{-ix_n \xi}\}\) form a Riesz basis for \(L_2 [-\pi, \pi]\). \textit{Carl de Boor} [Lect. Notes Math. 556, 30-53(1976; Zbl 0337.41004)] proved the existence of the interpolating spline \(S_kf\) under the condition of polynomial growth of \(f(x)\) as \(|x |\) approaches \(\infty\). The authors of this interesting paper prove the pointwise and \(L_2\) convergence of \(S_kf\) to \(f\) as \(k\) tends to \(\infty\) for every function \(f\) from the Paley-Wiener class \(PW_\pi\).