Abel Summability for a Distribution Sampling Theorem

Abstract

Let F(w)be an L² function with compact support on [-σ,σ]; let T = π/σ and f(t) be the Fourier transform of F(w). Then the well-known sampling theorem says

$${\text{f}}\left( {\text{t}} \right) = \sum\limits_{{\text{n}} = - \infty }^\infty {{\text{f}}\left( {{\text{nT}}} \right)\frac{{\sin \sigma \left( {{\text{t}} - {\text{nT}}} \right)}} {{\sigma \left( {{\text{t}} - {\text{nT}}} \right)}}},$$

where convergence is uniform in ℝ¹. If F(w) is now a distribution with compact support on [-σ,σ] the Fourier transform is still a function but the series does not converge necessarily. However it is shown, under mild conditions of F(w), that the series is Abel summable, i.e.

$${\text{f}}\left( {\text{t}} \right) = \mathop {\lim }\limits_{{\text{r}} \to l^ - } \sum\limits_{{\text{n}} = - \infty }^\infty {{\text{r}}^{\left| {\text{n}} \right|} \,{\text{f}}\left( {{\text{nT}}} \right)\frac{{\sin \sigma \left( {{\text{t}} - {\text{nT}}} \right)}} {{\sigma \left( {{\text{t}} - {\text{nT}}} \right)}}}$$

where the convergence is uniform on bounded sets in ℝ¹.