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Transactions of the American Mathematical Society

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The set of continuous functions with everywhere convergent Fourier series

Authors: M. Ajtai and A. S. Kechris
Journal: Trans. Amer. Math. Soc. 302 (1987), 207-221
MSC: Primary 04A15; Secondary 26A21, 42A20
MathSciNet review: 887506
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Abstract: This paper deals with the descriptive set theoretic properties of the class $ \operatorname{EC} $ of continuous functions with everywhere convergent Fourier series. It is shown that this set is a complete coanalytic set in $ C(T)$. A natural coanalytic rank function on $ \operatorname{EC} $ is studied that assigns to each $ f \in \operatorname{EC} $ a countable ordinal number, which measures the "complexity" of the convergence of the Fourier series of $ f$. It is shown that there exist functions in $ \operatorname{EC} $ (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on $ \operatorname{EC} $ with $ {\omega _1}$ distinct levels.

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