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Pathological Phenomena in Denjoy–Carleman Classes

Published online by Cambridge University Press:  20 November 2018

Ethan Y. Jaffe
Ethan Y. Jaffe*
Affiliation:
Massachusetts Institute of Technology, Department of Mathematics, Building E18, Room 369, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA e-mail: eyjaffe@mit.edu
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Abstract

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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 ${{C}^{M}}\left( {{\mathbb{R}}^{p}} \right)$.

Keywords

26E1026B3526E0530D6046E25Denjoy–Carleman classquasianalytic functionquasianalytic curvearc-quasianalytic
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 1 , 01 February 2016 , pp. 88 - 108
Copyright
Copyright © Canadian Mathematical Society 2016

References

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