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The Complexity of Everywhere Divergent Fourier Series

Published online by Cambridge University Press:  20 November 2018

T. I. Ramsamujh
T. I. Ramsamujh*
Affiliation:
Department of Mathematics Florida International University Miami, FL 33199, USA
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Abstract

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A natural rank function is defined on the set DS of everywhere divergent sequences of continuous functions on the unit circle T. The rank function provides a natural measure of the complexity of the sequences in DS, and is obtained by associating a well-founded tree with each such sequence. The set DF of everywhere divergent Fourier series, and the set DT of everywhere divergent trigonometric series with coefficients that tend to zero, can be viewed as natural subsets of DS. It is shown that the rank function is a coanalytic norm which is unbounded in ω1 on DF. From this it follows that DF, DT and DS are not Borel subsets of the Polish space SC(T) of all sequences of continuous functions on T. Finally an alternative definition of the rank function is formulated by using nested sequences of closed sets.

Keywords

04A1542A20everywhere divergentFourier seriesOrdinal rank
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 2 , 01 April 1991 , pp. 413 - 424
Copyright
Copyright © Canadian Mathematical Society 1991

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