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

 

One and two dimensional
Cantor-Lebesgue type theorems


Authors: J. Marshall Ash and Gang Wang
Journal: Trans. Amer. Math. Soc. 349 (1997), 1663-1674
MSC (1991): Primary 42A20, 42B99; Secondary 40A05, 40C99
MathSciNet review: 1357390
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Abstract: Let $\varphi (n)$ be any function which grows more slowly than exponentially in $n,$ i.e., $\mathop {limsup}\limits _{n\rightarrow \infty }\varphi (n)^{1/n}\leq 1.$ There is a double trigonometric series whose coefficients grow like $\varphi (n),$ and which is everywhere convergent in the square, restricted rectangular, and one-way iterative senses. Given any preassigned rate, there is a one dimensional trigonometric series whose coefficients grow at that rate, but which has an everywhere convergent partial sum subsequence. There is a one dimensional trigonometric series whose coefficients grow like $\varphi (n),$ and which has the everywhere convergent partial sum subsequence $S_{2^j}.$ For any $p>1,$ there is a one dimensional trigonometric series whose coefficients grow like $\varphi (n^{\frac {p-1}p}),$ and which has the everywhere convergent partial sum subsequence $S_{[j^p]}.$ All these examples exhibit, in a sense, the worst possible behavior. If $m_j$ is increasing and has arbitrarily large gaps, there is a one dimensional trigonometric series with unbounded coefficients which has the everywhere convergent partial sum subsequence $S_{m_j}.$


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Additional Information

J. Marshall Ash
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504
Email: mash@math.depaul.edu

Gang Wang
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504
Email: gwang@math.depaul.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01641-3
PII: S 0002-9947(97)01641-3
Keywords: Cantor-Lebesgue theorem, coefficient size, subsequences, trigonometric series, two dimensional trigonometric series, restricted rectangular convergence
Received by editor(s): February 23, 1994
Received by editor(s) in revised form: November 20, 1995
Additional Notes: J. M. Ash was partially supported by the National Science Foundation grant no. DMS-9307242. G. Wang was partially supported by grants from the Faculty Research and Development Program of the College of Liberal Arts and Sciences, DePaul University.
Article copyright: © Copyright 1997 American Mathematical Society