<abstract xmlns="http://www.w3.org/1999/xhtml">Given a <italic>φ</italic>-function <italic>φ</italic> and <italic>k</italic> ∈ ℕ, we introduce and study the concept of (<italic>φ, k</italic>)-variation in the sense of Riesz of a real function on a compact interval. We show that a function <italic>u</italic> :[<italic>a, b</italic>] <italic>→</italic> ℝ has a bounded (<italic>φ, k</italic>)-variation if and only if <italic>u</italic>(<italic>k−</italic>1) is absolutely continuous on [<italic>a, b</italic>]and <italic>u</italic>(<italic>k</italic>) belongs to the Orlicz class <italic>L φ</italic>[<italic>a, b</italic>]. 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 <italic>φ</italic>-variation, the de la Vallée Poussin second-variation and the Popoviciu <italic>k</italic>th variation.</abstract>

Given a φ-function φ and k ∈ 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] → 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 kth variation.

On Functions of Bounded (<italic>φ, k</italic>)-Variation

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":{"italic":"φ, k","content":["On Functions of Bounded (",")-Variation"]}}

Given a φ-function φ and k ∈ 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...

On Functions of Bounded (φ, k)-Variation

Published Online: Nov 15, 2019

Page range: 91 - 116

Received: Mar 03, 2016

DOI: https://doi.org/10.2478/tmmp-2019-0023

Keywords
Riesz -variation, De la Vallée Poussin second-variation, Popoviciu th variation, bounded ()-variation

© 2019 Hugo Leiva et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On Functions of Bounded (φ, k)-Variation

Published Online: Nov 15, 2019

Page range: 91 - 116

Received: Mar 03, 2016

DOI: https://doi.org/10.2478/tmmp-2019-0023

KeywordsRiesz -variation, De la Vallée Poussin second-variation, Popoviciu th variation, bounded ()-variation

© 2019 Hugo Leiva et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Riesz -variation, De la Vallée Poussin second-variation, Popoviciu th variation, bounded ()-variation