Open AccessOn Functions of Bounded (φ, k)-Variation
Author(s) -
Hugo Leiva,
Nelson Merentes,
Sergio Rivas,
José Luis Sánchez,
Małgorzata Wróbel
Publication year - 2019
Publication title -
tatra mountains mathematical publications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.171
H-Index - 12
eISSN - 1338-9750
pISSN - 1210-3195
DOI - 10.2478/tmmp-2019-0023
Subject(s) - mathematics , bounded variation , variation (astronomy) , bounded function , class (philosophy) , interval (graph theory) , pure mathematics , space (punctuation) , function (biology) , banach space , absolute continuity , mathematical analysis , combinatorics , linguistics , philosophy , physics , artificial intelligence , evolutionary biology , astrophysics , computer science , biology
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 k th variation.