Constructive approach to the monotone rearrangement of functions

• Published in: Expositiones mathematicae Vol. 40; no. 1; pp. 155 - 175
• Main Authors: Barbarino, Giovanni, Bianchi, Davide, Garoni, Carlo
• Format: Journal Article
• Language: English
• Published: Elsevier GmbH 01-03-2022
• Subjects: Almost everywhere continuous functions; Asymptotically uniform grids and quasi-uniform samples; Generalized inverse distribution function; Monotone rearrangement; Quantile function; Uniform convergence; secondary; Uniform convergence; Generalized inverse distribution function; Almost everywhere continuous functions; Monotone rearrangement; Asymptotically uniform grids and quasi-uniform samples; Quantile function; primary
• ISSN: 0723-0869; 1878-0792
• DOI: 10.1016/j.exmath.2021.10.004

We detail a simple procedure (easily convertible to an algorithm) for constructing, from quasi-uniform samples of f, a sequence of linear spline functions converging to the monotone rearrangement of f, in the case where f is an almost everywhere continuous function defined on a bounded set Ω with negligible boundary. Under additional assumptions on f and Ω, we prove that the convergence of the sequence is uniform. We also show that the same procedure applies to arbitrary measurable functions too, but with the substantial difference that in this case the procedure has only a theoretical interest and cannot be converted to an algorithm.