The paper studies spectral representation and its applications for non-decaying continuous time signals that are not necessarily bounded at $\pm\infty$. The paper introduces notions of transfer functions, spectrum degeneracy, spectrum gaps, and bandlimitness, for these unbounded signals. As an example of applications, explicit formulae are given for transfer functions of low-pass and high-pass filters suitable for these signal. As another example of applications, it is shown that non-decaying unbounded signals with a single point spectrum degeneracy and sublinear rate of growth are predictable. The corresponding transfer functions for the predictors are obtained explicitly.