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# Pathological Phenomena in Denjoy-Carleman Classes

Published:2015-05-21
Printed: Feb 2016
• Ethan Y. Jaffe,
University of Toronto, Department of Mathematics, 40 St. George Street, Toronto, ON M5S 2E4
 Format: LaTeX MathJax PDF

## 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 $\mathcal{C}^M(\mathbb{R}^p)$.
 Keywords: Denjoy-Carleman classes, quasianalytic functions, quasianalytic curve, arc-quasianalytic
 MSC Classifications: 26E10 - $C^\infty$-functions, quasi-analytic functions [See also 58C25]

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