  Suppose that a sequence of numbers $x_n$ (a `signal') is transmitted through
a noisy channel. The receiver observes a noisy version of the signal with
additive random fluctuations, $x_n + \xi_n$, where $\xi_n$ is a sequence of
independent standard Gaussian random variables. Suppose further that the signal
is known to come from some fixed space of possible signals. Is it possible to
fully recover the transmitted signal from its noisy version? Is it possible to
at least detect that a non-zero signal was transmitted?
  In this paper we consider the case in which signals are infinite sequences
and the recovery or detection are required to hold with probability one. We
provide conditions on the signal space for checking whether detection or
recovery are possible. We also analyze in detail several examples including
spaces of Fourier transforms of measures, spaces with fixed amplitudes and the
space of almost periodic functions. Many of our examples exhibit critical
phenomena, in which a sharp transition is made from a regime in which recovery
is possible to a regime in which even detection is impossible.
