A new inequality for the Riemann-Stieltjes integrals driven by irregular signals in Banach spaces

Autor: Rafał M Łochowski
Jazyk: angličtina
Rok vydání: 2018
Předmět:
Zdroj: Journal of Inequalities and Applications, Vol 2018, Iss 1, Pp 1-23 (2018)
Druh dokumentu: article
ISSN: 1029-242X
DOI: 10.1186/s13660-018-1611-4
Popis: 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 ≥ 1 $p\ge1$ , we introduce the space of regulated signals f : [ a , b ] → W $f:[a,b]\rightarrow W$ ( a < b $a< b$ are real numbers, and W is a Banach space) that may be uniformly approximated with accuracy δ > 0 $\delta>0$ by signals whose total variation is of order δ 1 − p $\delta^{1-p}$ as δ → 0 + $\delta\rightarrow0+$ 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.
Databáze: Directory of Open Access Journals
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