Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Localization properties of basic classes of $ C\sp{\infty }$-functions

Author: R. B. Darst
Journal: Proc. Amer. Math. Soc. 46 (1974), 24-28
MSC: Primary 26A93
MathSciNet review: 0346113
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that each of $ \mathfrak{M} = C\{ {M_n}\} $ and $ \mathcal{K} = C\{ {K_n}\} $ is a basic class of $ {C^\infty }$-functions defined on $ I = [0,2\pi ]$ and $ \mathfrak{M}$ is not a subset of $ \mathcal{K}$. Then it is shown that $ \mathfrak{M}$ contains functions which are nowhere locally in $ \mathcal{K}$. One of the corollaries asserts that there are quasi-analytic functions which are nowhere locally analytic. It is also shown that many non-quasi-analytic classes contain functions which are nowhere locally quasi-analytic.

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

  • [1] A. Gorny, Contribution à l’étude des fonctions dérivables d’une variable réelle, Acta Math. 71 (1939), 317–358 (French). MR 0000848
  • [2] Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
  • [3] S. Mandelbrojt, Analytic functions and classes of infinitely differentiable functions, Rice Inst. Pamphlet 29 (1942), no. 1, 142. MR 0006354
  • [4] L. E. May, On 𝐶^{∞} functions analytic on sets of small measure, Canad. Math. Bull 12 (1969), 25–30. MR 0241594
  • [5] Walter Rudin, Principles of mathematical analysis, Second edition, McGraw-Hill Book Co., New York, 1964. MR 0166310
  • [6] Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A93

Retrieve articles in all journals with MSC: 26A93

Additional Information

Keywords: Analytic functions, $ {C^\infty }$-functions, quasi-analytic functions
Article copyright: © Copyright 1974 American Mathematical Society