On predictors and filters for non-decaying unbounded continuous time signals

TL;DR

The paper develops a dual spectral framework for non-decaying unbounded continuous-time signals by introducing the spaces ρ^{-1}L_∞ and 𝔅(ρ) and a Fourier-based mapping ${\cal F}: ρ^{-1}L_∞ \to 𝔅(ρ)^*$. It defines transfer functions, spectrum degeneracy, and band-limitness in this setting and constructs practical low-pass/high-pass filters alongside explicit predictors for signals with a single point spectrum degeneracy (Theorem ThP). The results include a Parseval-type identity, a bijection between ${\cal B}(ρ)^*$ and ρ^{-1}L_∞, and convergence analyses showing that past history uniquely determines the future under certain degeneracy, thereby extending data-recovery and prediction theory to unbounded signals. The work provides concrete filtering and predictive tools for signals with polynomial growth, with implications for signal processing where standard decaying assumptions fail, and outlines open questions on density of band-limited signals and alternative weight functions $ρ$.

Abstract

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.

Theorems & Definitions (24)

• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• Proposition 2.4
• Remark 2.5
• Proposition 2.6
• Theorem 2.7
• Corollary 2.8
• Definition 3.1
• Example 3.2