Asymptotically uniform functions: a single hypothesis which solves two old problems

2023/01/25 by Gabriel, Jean-Pierre, Berrut, Jean-Paul
#26A09 #26A12 #34A45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2301.10505

Abstract

The asymptotic study of a time-dependent function f as the solution of a differential equation often leads to the question of whether its derivative f vanishes at infinity. We show that a necessary and sufficient condition for this is that f is what may be called "asymptotically uniform". We generalize the result to higher order derivatives. We further show that the same property for f itself is also necessary and sufficient for its one-sided improper integrals to exist. On the way, the article provides a broad study of such asymptotically uniform functions.

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