Pathological phenomena in Denjoy-Carleman classes

Abstract

Let ${\mathcal{C}}^M$ denote a Denjoy-Carleman class of ${\mathcal{C}}^{\infty }$ functions (for a given logarithmically-convex sequence $M=(M_n)$). We construct: (1) a function in ${\mathcal{C}}^M((-1,1))$ which is nowhere in any smaller class; (2) a function on $\mathbb{R}$ which is formally ${\mathcal{C}}^M$ at every point, but not in ${\mathcal{C}}^M(\mathbb{R})$; (3) (under the assumption of quasianalyticity) a smooth function on ${\mathbb{R}}^p$ ($p\ge 2$) which is ${\mathcal{C}}^M$ on every ${\mathcal{C}}^M$ curve, but not in ${\mathcal{C}}^M({\mathbb{R}}^p)$.

21 pages

References in corpus (1)

• Failure of the Weierstrass Preparation Theorem in quasi-analytic Denjoy-Carleman rings

Cited by in corpus (4)

• Arc-smooth functions on closed sets
• Ultradifferentiable extension theorems: a survey
• Nonlinear conditions for ultradifferentiability
• Composite quasianalytic functions