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Transactions of the American Mathematical Society

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On cubic lacunary Fourier series

Author: Joseph L. Gerver
Journal: Trans. Amer. Math. Soc. 355 (2003), 4297-4347
MSC (2000): Primary 42A55, 26A27
Published electronically: July 2, 2003
MathSciNet review: 1990754
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Abstract: For $2<\beta <4$, we analyze the behavior, near the rational points $x=p\pi /q$, of $\sum^\infty_{n=1}n^{-\beta }\exp (ixn^{3})$, considered as a function of $x$. We expand this series into a constant term, a term on the order of $(x-p\pi /q)^{(\beta -1)/3}$, a term linear in $x-p\pi /q$, a ``chirp" term on the order of $(x-p\pi /q)^{(2\beta -1)/4}$, and an error term on the order of $(x-p\pi /q)^{\beta /2}$. At every such rational point, the left and right derivatives are either both finite (and equal) or both infinite, in contrast with the quadratic series, where the derivative is often finite on one side and infinite on the other. However, in the cubic series, again in contrast with the quadratic case, the chirp term generally has a different set of frequencies and amplitudes on the right and left sides. Finally, we show that almost every irrational point can be closely approximated, in a suitable Diophantine sense, by rational points where the cubic series has an infinite derivative. This implies that when $\beta \le (\sqrt {97}-1)/4=2.212\dots $, both the real and imaginary parts of the cubic series are differentiable almost nowhere.

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Additional Information

Joseph L. Gerver
Affiliation: Department of Mathematics, Rutgers University, Camden, New Jersey 08102

Received by editor(s): October 18, 1999
Published electronically: July 2, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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