Pathological Phenomena in Denjoy–Carleman Classes

Abstract

AbstractLet M denote a Denjoy–Carleman class of ∞ functions (for a given logarithmically-convex sequence M = (Mn)). We construct: (1) a function in M((−1, 1)) that is nowhere in any smaller class; (2) a function on ℝ that is formally M at every point, but not in M (ℝ); (3) (under the assumption of quasianalyticity) a smooth function on ℝp (p ≥ 2) that is M on every M curve, but not inM (ℝp).