Sampling Theorem and interpolation formula for non-vanishing signals

TL;DR

The paper addresses recovering non-vanishing continuous signals from samples by deriving an analog of the Nyquist–Shannon theorem and a modified Whittaker–Shannon–Kotelnikov interpolation formula. It constructs a time-varying interpolation kernel via a function $g(t)$ and parameter choices to yield an exact expansion $x(t)=\sum_k a_k(t)x(k)$ for $x\in C(\mathbb{R})\cap \mathcal{V}(\Omega)$, with coefficients decaying as $|a_k(t)|\sim 1/k^{2}$. The work introduces a spectral-representation framework for non-vanishing signals, relating them to band-limited theory through a dual space mapping $\mathcal{F}: L_{\infty} \to {\cal A}^{*}$ and associated convolution operators. This yields a practical, exact interpolation method that remains robust for non-vanishing signals and enables improved numerical accuracy in sample-based reconstruction, even when traditional $1/k$ decay makes the classical formula inapplicable.

Abstract

The paper establishes an analog Whittaker-Shannon-Kotelnikov sampling theorem with fast decreasing coefficient, as well as a new modification of the corresponding interpolation formula applicable for general type non-vanishing bounded continuous signals.

Theorems & Definitions (17)

• Definition 2.1
• Theorem 2.2
• Corollary 2.3
• Lemma 4.1
• Lemma 4.2
• Proposition 4.3
• Remark 4.4
• Lemma 4.5
• Theorem 4.6
• Lemma 4.7