The paper studies spectral representation as well as predictability and recoverability problems for non-vanishing discrete time signals from $\ell_\infty$, i.e. for bounded discrete time signals, including signals that do not vanish at $\pm\infty$. The extends the notions of transfer functions, the spectrum gaps, bandlimitness, and filters, on these general type signals. Some frequency conditions of predictability and data recoverability are presented, and some recovery methods and predictors have been suggested.