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

Abstract

We prove an inequality of the Loéve-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 $p\ge 1$ we introduce the space of regulated signals $f:[a,b]\to W$ ($a<b$ 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 $\delta \to 0+$ 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.

arXiv admin note: text overlap with arXiv:1409.3757

References in corpus (2)

• The Play Operator, the Truncated Variation and the Generalisation of the Jordan Decomposition
• A new theorem on the existence of the Riemann-Stieltjes integral and an improved version of the Loéve-Young inequality

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