On functions of bounded $(φ, k)$-variation

Year, pages: 2019, 91 - 116

Given a $φ $-function $φ $ and $k\in \mathbb{N}$, we introduce and study the concept of $(φ ,k)$-variation in the sense of Riesz of a real function on a compact interval. We show that a function $u :[a,b]\rightarrow \mathbb{R}$ has a bounded $(φ, k)$-variation if and only if $u^(k-1)$ is absolutely continuous on $[a,b]$ and $u^(k)$ belongs to the Orlicz class $L_φ[a,b]$. We also show that the space generated by this class of functions is a Banach space. Our approach simultaneously generalizes the concepts of the Riesz $φ $-variation, the de la Vallée Poussin second-variation and the Popoviciu $k$th variation.

Leiva, H., Merentes, N., Rivas, S., Sánchez, J., Wróbel, M. 2019. On functions of bounded $(φ, k)$-variation. In Tatra Mountains Mathematical Publications, vol. 74, no.2, pp. 91-116. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2019-0023

Leiva, H., Merentes, N., Rivas, S., Sánchez, J., Wróbel, M. (2019). On functions of bounded $(φ, k)$-variation. Tatra Mountains Mathematical Publications, 74(2), 91-116. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2019-0023

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava

Licensed under the Creative Commons Attribution-NC-ND4.0 International Public License.