Pathological phenomena in Denjoy-Carleman classes

TL;DR

This paper explores complex pathological phenomena in Denjoy-Carleman classes, constructing functions with unusual smoothness properties that challenge traditional notions of function regularity and class membership.

Contribution

It introduces new examples of functions exhibiting pathological behaviors within Denjoy-Carleman classes, including functions that are nowhere in smaller classes and functions that are pointwise but not globally in the class.

Findings

Constructed functions that are in a class but nowhere in smaller classes

Functions formally $ ext{C}^M$ at every point but not globally in $ ext{C}^M$

Existence of smooth functions on $ ext{R}^p$ that are $ ext{C}^M$ on all curves but not in $ ext{C}^M( ext{R}^p)$

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 $\mathbb R^p$ ($p \geq 2$) which is $\mathcal C^M$ on every $\mathcal C^M$ curve, but not in $\mathcal C^M(\mathbb R^p)$.