Power substitution in quasianalytic Carleman classes

TL;DR

This paper investigates how solutions to specific functional equations involving powers preserve quasianalytic properties within Carleman classes, extending results to multivariate functions.

Contribution

It constructs new Carleman classes containing solutions to power substitution equations and proves quasianalyticity preservation under certain conditions.

Findings

New Carleman classes for solutions are constructed

Quasianalyticity is preserved in the new classes

Results extend to multivariate functions

Abstract

Consider an equation of the form $f(x)=g(x^k)$, where $k>1$ and $f(x)$ is a function in a given Carleman class of smooth functions. For each $k$, we construct a Carleman-type class which contains all the smooth solutions $g(x)$ to such equations. We prove, under regularity assumptions, that if the original Carleman class is quasianalytic, then so is the new class. The results admit an extension to multivariate functions.