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Published July 1987 | Published
Journal Article Open

The Set of Continuous Functions with the Everywhere Convergent Fourier Series

Abstract

This paper deals with the descriptive set theoretic properties of the class 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 EC is studied that assigns to each ƒ Є EC a countable ordinal number, which measures the "complexity" of the convergence of the Fourier series of ƒ. It is shown that there exist functions in EC (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on EC with ω_1 distinct levels.

Additional Information

© 1987 American Mathematical Society. Received by the editors June 11, 1985 and, in revised form, June 20, 1986. Research partially supported by NSF Grant DMS-8416349.

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