Discrete time scattering and Wold's decomposition in pictures

TL;DR

This paper reviews and illustrates discrete time scattering theory concepts on a half-space, providing explicit formulas and visualizations to clarify the interpretation of wave operators and Wold's decomposition.

Contribution

It offers a pedagogical presentation with explicit formulas and graphical representations of operators in discrete time scattering theory and Wold's decomposition.

Findings

Explicit formulas for wave operators

Graphical representations of operators

Visual interpretation of Wold's decomposition

Abstract

Based on explicit computations, various concepts of discrete time scattering theory are reviewed, discussed, and illustrated. The dynamics are taking place on a discrete half-space. All operators are represented graphically. The expressions obtained for the wave operators lead to an easily visualized interpretation of Wold's decomposition, a seminal result of operator theory. This work is clearly pedagogically oriented with the aim of providing explicit formulas and pictures for usually unexplicit operators.