Abstract view
Pathological Phenomena in DenjoyCarleman Classes


Published:20150521
Printed: Feb 2016
Ethan Y. Jaffe,
University of Toronto, Department of Mathematics, 40 St. George Street, Toronto, ON M5S 2E4
Abstract
Let $\mathcal{C}^M$ denote a DenjoyCarleman class of $\mathcal{C}^\infty$
functions (for a given logarithmicallyconvex 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 $\mathcal{C}^M(\mathbb{R}^p)$.