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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Kechris-Woodin rank is finer than the Zalcwasser rank


Author: Haseo Ki
Journal: Trans. Amer. Math. Soc. 347 (1995), 4471-4484
MSC: Primary 04A15; Secondary 26A21, 26A24, 42A20
MathSciNet review: 1321581
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Abstract: For each differentiable function $ f$ on the unit circle, the Kechris-Woodin rank measures the failure of continuity of the derivative function $ f'$ while the Zalcwasser rank measures how close the Fourier series of $ f$ is to being a uniformly convergent series. We show that the Kechris-Woodin rank is finer than the Zalcwasser rank. Roughly speaking, small ranks mean the function is well behaved and big ranks imply bad behavior. For each countable ordinal, we explicitly construct a continuous function with everywhere convergent Fourier series such that the Zalcwasser rank of the function is bigger than the ordinal.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1321581-2
PII: S 0002-9947(1995)1321581-2
Keywords: Denjoy rank, descriptive set theory, Fourier series, Kechris-Woodin rank, Zalcwasser rank
Article copyright: © Copyright 1995 American Mathematical Society