On the Borel mapping in the quasianalytic setting

TL;DR

This paper proves that the Borel mapping cannot be surjective for quasianalytic ultradifferentiable classes larger than real analytic functions, highlighting limitations in reconstructing functions from their derivatives.

Contribution

It establishes a non-surjectivity result for the Borel mapping in quasianalytic ultradifferentiable classes larger than analytic functions.

Findings

Borel mapping is not onto for these classes

Limits of reconstructing functions from derivatives in quasianalytic classes

Extension of non-surjectivity beyond real analytic functions

Abstract

The Borel mapping takes germs at $0$ of smooth functions to the sequence of iterated partial derivatives at $0$. We prove that the Borel mapping restricted to the germs of any quasianalytic ultradifferentiable class strictly larger than the real analytic class is never onto the corresponding sequence space.