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Ultradifferentiable functions on lines in $\mathbb{R}^{n}$


Author: Tejinder Neelon
Journal: Proc. Amer. Math. Soc. 127 (1999), 2099-2104
MSC (1991): Primary 30D60; Secondary 46F05
DOI: https://doi.org/10.1090/S0002-9939-99-04759-0
Published electronically: March 16, 1999
Erratum: Proc. Amer. Math. Soc. 131 (2003), 991-992.
MathSciNet review: 1487332
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Abstract: It is well known that a function $f\in C^{\infty }(\mathbb{R}^{n})$ whose restriction to every line in $\mathbb{R}^{n}$ is real analytic must itself be real analytic. In this note we study whether this property of real analytic functions is also possessed by some other subclasses of $C^{\infty } $ functions. We prove that if $f\in C^{\infty }(\mathbb{R}^{n})$ is ultradifferentiable corresponding to a sequence $\{M_{k}\}$ on every line in some `uniform way', then $f$ is ultradifferentiable corresponding to the sequence $\{M_{k}\}.$


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Additional Information

Tejinder Neelon
Affiliation: Department of Mathematics, California State University San Marcos, San Marcos, California 92096-0001
Email: NEELON@MAILHOST1.CSUSM.EDU

DOI: https://doi.org/10.1090/S0002-9939-99-04759-0
Keywords: Ultradifferentiable functions, Vandermonde determinants
Received by editor(s): August 28, 1997
Received by editor(s) in revised form: October 15, 1997
Published electronically: March 16, 1999
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society