On the Equivalence Between Classical and Distributional Convergence for Shannon Type Interpolation Series and Applications

Abstract:
The distribution theory serves as an important theoretical foundation for some approaches arose from the engineering intuition. Particular examples are approaches based on the δ-”function”. We show that for the Shannon sampling/ interpolation series (SSS/SIS) of continuous signals ”vanishing” at infinity, the classical notion of convergence given in complex analysis is equivalent with the modern notion given by the distribution theory, in the sense that the SSS converges at a point on the real line, different from the sampling/interpolation point, if and only if it converges distributionally. This result is in spirit of Weyl’s Lemma on the Laplace equation. As an extension, we give those results also for the sampling/interpolation series based on the sine-type function.