Pathological Phenomena in Denjoy–Carleman Classes

Jaffe, Ethan Y.

Canadian journal of mathematics, Tome 68 (2016) no. 1, pp. 88-108

Résumé

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

Détail

DOI : 10.4153/CJM-2015-009-3

Mots-clés : 26E10, 26B35, 26E05, 30D60, 46E25, Denjoy–Carleman class, quasianalytic function, quasianalytic curve, arc-quasianalytic