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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Zero sets of functions from non-quasi-analytic classes


Author: R. B. Darst
Journal: Proc. Amer. Math. Soc. 40 (1973), 543-544
MSC: Primary 26A93
DOI: https://doi.org/10.1090/S0002-9939-1973-0323978-7
MathSciNet review: 0323978
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Abstract: Any closed subset $ E$ of the real numbers $ R$ is the zero set of some $ {C^\infty }$-function $ f$. One can also specify the order $ d(s)$ of the zero of $ f$ at each element $ s$ of the set $ S$ of isolated points of $ E$. The present note improves this result by showing that each non-quasi-analytic class $ C\{ {M_n}\} $ contains such functions.


References [Enhancements On Off] (What's this?)

  • [1] R. B. Hughes, Zero sets of functions from non-quasi-analytic classes, Proc. Amer. Math. Soc. 27 (1971), 539-542. MR 0272965 (42:7846)
  • [2] S. Mandelbrojt, Analytic functions and classes of infinitely differentiable functions, Rice Inst. Pamphlet 29, no. 1 (1942). MR 3, 292. MR 0006354 (3:292d)
  • [3] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. MR 35 #1420. MR 0210528 (35:1420)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0323978-7
Keywords: $ {C^\infty }$-functions, non-quasi-analytic classes, zero sets
Article copyright: © Copyright 1973 American Mathematical Society

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