Ultradifferentiable functions on lines in $\mathbb {R}^n$

Abstract:

It is well known that a function $f\in C^{\infty }(\mathbb {R}^{n})$ whose restriction to every line in $\mathbb {R}^{n}$ is real analytic must itself be real analytic. In this note we study whether this property of real analytic functions is also possessed by some other subclasses of $C^{\infty }$ functions. We prove that if $f\in C^{\infty }(\mathbb {R}^{n})$ is ultradifferentiable corresponding to a sequence $\{M_{k}\}$ on every line in some ‘uniform way’, then $f$ is ultradifferentiable corresponding to the sequence $\{M_{k}\}.$