On ultradifferentiable functions

TL;DR

This paper characterizes spaces of ultradifferentiable functions closed under composition with holomorphic or ultradifferentiable functions, using a power series approach that simplifies regularity proofs for solutions of ODEs and PDEs.

Contribution

It provides complete descriptions of ultradifferentiable function spaces closed under composition, avoiding differentiation stability requirements and simplifying regularity proofs.

Findings

Complete descriptions of ultradifferentiable function spaces under composition

Reproves regularity results for ODEs and PDEs without complex estimates

Uses power series approach to simplify proofs

Abstract

We give complete and exact descriptions of spaces of ultradifferentiable functions that are closed under composition with either holomorphic or ultradifferentiable functions -- which are two distinct cases. The proof works by considering formal power series, and stability under differentiation is not required. As an application of the power series approach we reprove regularity results for solutions of ode's and pde's without employing tedious estimates imploying the Fa\`a di Bruno formula for higher derivatives of composite maps.