On functions of bounded variation
Provides foundational theoretical results for multivariate variation theory, benefiting mathematicians working on integration and discrepancy.
The paper resolves an open question by proving that every function of bounded Hardy-Krause variation is Borel measurable and has bounded D-variation, and shows that the space of such functions forms a commutative Banach algebra.
The recently introduced concept of $\mathcal{D}$-variation unifies previous concepts of variation of multivariate functions. In this paper, we give an affirmative answer to the open question from Pausinger \& Svane (J. Complexity, 2014) whether every function of bounded Hardy--Krause variation is Borel measurable and has bounded $\mathcal{D}$-variation. Moreover, we show that the space of functions of bounded $\mathcal{D}$-variation can be turned into a commutative Banach algebra.