Riemann-Stieltjes integrals driven by irregular signals in Banach spaces and rate-independent characteristics of their irregularity

TL;DR

This paper establishes new inequalities and existence results for Riemann-Stieltjes integrals driven by irregular signals in Banach spaces, introducing a regulated signals space and characterizing their irregularity in a rate-independent manner.

Contribution

It introduces a Loéve-Young type inequality, a new existence theorem for such integrals, and a novel space of regulated signals with rate-independent irregularity characterization.

Findings

Proved a Loéve-Young type inequality for Banach space-valued signals.

Established conditions for the existence of Riemann-Stieltjes integrals driven by irregular signals.

Characterized the irregularity of integrals in a rate-independent framework.

Abstract

We prove an inequality of the Lo\'{e}ve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any $p\ge1$ we introduce the space of regulated signals $f:[a,b] \rightarrow W$ ($a<b$ are real numbers and $W$ is a Banach space), which may be uniformly approximated with accuracy $\delta>0$ by signals whose total variation is of order $\delta^{1-p}$ as $\delta\rightarrow 0+$ and prove that they satisfy the assumptions of the theorem. Finally, we derive more exact, rate-independent characterisations of the irregularity of the integrals driven by such signals.