We analyze the problem of reconstruction of a bandlimited function f from the space–time samples of its states \(f_t=\phi _t*f\) resulting from the convolution with a kernel \(\phi _t\). It is well-known that, in natural phenomena, uniform space–time samples of f are not sufficient to reconstruct f in a stable way. To enable stable reconstruction, a space–time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space–time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function \(\widehat{f}\) away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Turán type inequalities. En route, we obtain Remez-Turán inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.

我们分析一个带限功能的重建问题˚F从其状态的空间时间样本\（F_T = \披_t * F \）从与内核的卷积所得\（\披_t \） 。众所周知，在自然现象中，均匀的f时空样本不足以重建f以稳定的方式。为了实现稳定的重建，必须使用Lu和Vetterli所展示的具有周期性不均匀间隔采样的时空采样。我们表明，重建的稳定性（由条件编号衡量）控制了空间样本之间的最大差距。我们提供此结果的定量说明。此外，代替了不规则的时空样本，我们证明了以亚奈奎斯特空间速率的均匀动力学样本可以使人稳定地重建函数\（\ widehat {f} \）远离某些明确描述的盲点。我们还考虑了带限制函数的几类有限维子集，即使在盲点内部，也可以进行稳定的重构。我们使用Remez-Turán型不等式获得定量估计。沿途，我们获得了球面扁长波函数的Remez-Turán不等式。为了说明我们的结果，我们提供一些有关热流问题的数值和显式估计。