On regularity for de Rham's functional equations

TL;DR

This paper investigates the regularity of solutions to de Rham's functional equations, extending previous results to non-linear functions like polynomials and fractional transformations, and analyzing the singularity of well-known functions such as Minkowski's question-mark function.

Contribution

It improves existing regularity results for de Rham's equations under smoothness conditions and applies to non-linear functions, revealing singularity properties of classical functions.

Findings

Solutions exhibit singularity in specific cases.

Results extend to non-linear functions like polynomials.

Implications for Minkowski's question-mark function.

Abstract

We consider regularity for solutions of a class of de Rham's functional equations. Under some smoothness conditions of functions consisting the equation, we improve some results in Hata (Japan J. Appl. Math. 1985). Our results are applicable to some cases that the functions consisting the equation are non-linear functions on an interval, specifically, polynomials and linear fractional transformations. Our results imply singularity of some well-known singular functions, in particular, Minkowski's question-mark function, and, some small perturbed functions of the singular functions.