Separating signal from noise

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 ${\mathscr {X}}$ 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 1. We provide conditions on the space ${\mathscr {X}}$ 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.