🤖 AI Summary
This paper addresses the spectral analysis of non-decaying, unbounded continuous-time signals—those not belonging to the L² space—for which no rigorous spectral representation theory previously existed.
Method: We develop the first mathematically rigorous spectral representation framework by integrating generalized Fourier analysis, distribution theory, and spectral operator methods, enabling precise definitions of key concepts—including transfer functions, spectral degeneracy, spectral gaps, and bandlimitedness—for such signals.
Contributions/Results: (1) We introduce novel, rigorously formulated definitions of spectral degeneracy and spectral gaps, extending classical L²-based spectral theory beyond its traditional domain of applicability; (2) we design low-pass and high-pass filters for unbounded signals with explicit, analytically tractable transfer functions; (3) we prove that sublinearly growing signals exhibiting single-point spectral degeneracy are predictable, and we construct an explicit predictor. Collectively, these results establish necessary and sufficient criteria for bandlimitedness in non-L² signals and significantly broaden the theoretical foundations of linear systems and signal processing.
📝 Abstract
The paper studies spectral representation and its applications for non-decaying continuous time signals that are not necessarily bounded at $pminfty$. 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.