We study spectral representation and its applications for non-decaying continuous-time signals that are not necessarily bounded at $\pm\infty$ . We introduce the notions of transfer functions, spectrum degeneracy, spectrum gaps, and band-limitedness, for these unbounded signals. As an example of applications, we give explicit formulas for transfer functions of low-pass and high-pass filters suitable for these signals. As another example of applications, we show that non-decaying unbounded signals with a single point spectrum degeneracy and polynomial rate of growth are predictable. The corresponding transfer functions for the predictors are obtained explicitly. In addition, we obtain an interpolation sampling formula for unbounded bandlimited signals.
