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On the set of all continuous functions
with uniformly convergent Fourier series

Author: Haseo Ki
Journal: Proc. Amer. Math. Soc. 124 (1996), 3507-3514
MSC (1991): Primary 04A15, 26A21; Secondary 42A20
MathSciNet review: 1340391
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Abstract: In this article we calculate the exact location in the Borel hierarchy of $UCF,$ the set of all continuous functions on the unit circle with uniformly convergent Fourier series. It turns out to be complete $F_{\sigma \delta }.$ Also we prove that any $G_{\delta \sigma }$ set that includes $UCF$ must contain a continuous function with divergent Fourier series.

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

  • [AK] M. Ajtai and A. S. Kechris, The set of continuous functions with everywhere convergent Fourier series, Trans. Amer. Math. Soc. 302 (1987), 207--221. MR 89b:04005
  • [Ka] Y. Katznelson, An introduction to harmonic analysis, Dover, New York, 1976. MR 54:10976
  • [Ke] Alexander S. Kechris, Classical Descriptive Set Theory, Springer Verlag, 1995. CMP 95:09
  • [KL] Haseo Ki and Tom Linton, Normal numbers and subsets of ${\mathbb N} $ with given densities, Fundamenta Mathematicae (2) 144 (1994), 163--179. MR 95e:04005
  • [Zy] A. Zygmund, Trigonometric series, 2nd ed., Cambridge Univ. Press, 1959. MR 21:6498

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Additional Information

Haseo Ki
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Address at time of publication: GARC, Department of Mathematics, Seoul National University, Seoul 151-742, Korea

Keywords: Descriptive set theory, Fourier series, complete $F_{\sigma \delta }$, uniformly convergent Fourier series
Received by editor(s): May 26, 1994
Received by editor(s) in revised form: May 12, 1995
Additional Notes: The author was partially supported by GARC-KOSEF
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1996 American Mathematical Society

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