Predictability of band-limited, high-frequency, and mixed processes in the presence of ideal low-pass filters

TL;DR

This paper investigates the predictability of various continuous-time processes, demonstrating that band-limited and high-frequency processes, as well as mixed processes with ideal low-pass filtering, can be predicted in a deterministic setting.

Contribution

It establishes the conditions under which band-limited, high-frequency, and mixed processes are predictable using ideal low-pass filters, extending previous understanding of process predictability.

Findings

Band-limited processes are predictable.

High-frequency processes with zero low-frequency energy are predictable.

Mixed processes can be predicted if an ideal low-pass filter exists.

Abstract

Pathwise predictability of continuous time processes is studied in deterministic setting. We discuss uniform prediction in some weak sense with respect to certain classes of inputs. More precisely, we study possibility of approximation of convolution integrals over future time by integrals over past time. We found that all band-limited processes are predictable in this sense, as well as high-frequency processes with zero energy at low frequencies. It follows that a process of mixed type still can be predicted if an ideal low-pass filter exists for this process.