The conversion of analog signals into digital signals and vice versa, performed by sampling and interpolation, respectively, is an essential operation in signal processing. When digital computers are used to compute the analog signals, it is important to effectively control the approximation error. In this paper we analyze the computability, i.e., the effective approximation of bandlimited signals in the Bernstein spaces $\mathcal {B}_{\pi }^p$, $1 \leq p < \infty$, and of the corresponding discrete-time signals that are obtained by sampling. We show that for $1< p< \infty$, computability of the discrete-time signal implies computability of the continuous-time signal. For $p=1$ this correspondence no longer holds. Further, we give a necessary and sufficient condition for computability and show that the Shannon sampling series provides a canonical approximation algorithm for $p>1$. We discuss BIBO stable LTI systems and the time-domain concentration behavior of bandlimited signals as applications.

中文翻译：

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图灵遇上香农：具有误差控制的可计算采样类型重构

模拟信号与数字信号的相互转换，分别通过采样和插值进行，是信号处理中的基本操作。当使用数字计算机计算模拟信号时，有效控制逼近误差很重要。在本文中，我们分析了可计算性，即伯恩斯坦空间中带限信号的有效逼近$\mathcal {B}_{\pi }^p$, $1 \leq p < \infty$，以及通过采样获得的相应离散时间信号。我们证明对于$1< p< \infty$，离散时间信号的可计算性意味着连续时间信号的可计算性。为了$p=1$这封信不再成立。此外，我们给出了可计算性的充分必要条件，并表明香农采样序列为$p>1$. 我们讨论了 BIBO 稳定 LTI 系统和作为应用的带限信号的时域集中行为。