On the Discrepancy Normed Space of Event Sequences for Threshold-based Sampling

TL;DR

This paper explores the mathematical structure of event sequences under threshold-based sampling, introducing a discrepancy normed space and revealing properties like Jordan decomposition and links to total variation.

Contribution

It extends the space of event sequences to a normed space with Hermann Weyl's discrepancy measure, providing new insights into their metric and topological properties.

Findings

Sequences with finite discrepancy norm have Jordan decomposition

Dual norm corresponds to total variation norm

An inequality akin to Heisenberg's uncertainty relation is established

Abstract

Recalling recent results on the characterization of threshold-based sampling as quasi-isometric mapping, mathematical implications on the metric and topological structure of the space of event sequences are derived. In this context, the space of event sequences is extended to a normed space equipped with Hermann Weyl's discrepancy measure. Sequences of finite discrepancy norm are characterized by a Jordan decomposition property. Its dual norm turns out to be the norm of total variation. As a by-product a measure for the lack of monotonicity of sequences is obtained. A further result refers to an inequality between the discrepancy norm and total variation which resembles Heisenberg's uncertainty relation.