  We analyze the problem of reconstruction of a bandlimited function $f$ from
the space-time samples of its states $f_t=\phi_t\ast f$ 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-Tur\'an type inequalities. En route,
we obtain a Remez-Tur\'an inequality for prolate spheroidal wave functions. To
illustrate our results, we present some numerics and explicit estimates for the
heat flow problem.
