On ultradifferentiable functions
J\"urgen P\"oschel

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Advanced Topics in Algebra · Holomorphic and Operator Theory
