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 | References | Similar Articles | Additional Information

Abstract: Let be any function which grows more slowly than exponentially in i.e., There is a double trigonometric series whose coefficients grow like 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 and which has the everywhere convergent partial sum subsequence For any there is a one dimensional trigonometric series whose coefficients grow like and which has the everywhere convergent partial sum subsequence All these examples exhibit, in a sense, the worst possible behavior. If 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

**[AW]**J. Marshall Ash and Grant V. Welland,*Convergence, uniqueness, and summability of multiple trigonometric series*, Trans. Amer. Math. Soc.**163**(1972), 401–436. MR**0300009**, 10.1090/S0002-9947-1972-0300009-X**[AFR]**J. Marshall Ash, Chris Freiling, and Dan Rinne,*Uniqueness of rectangularly convergent trigonometric series*, Ann. of Math. (2)**137**(1993), no. 1, 145–166. MR**1200079**, 10.2307/2946621**[AKR]**J. Marshall Ash, Eric Rieders, and Robert P. Kaufman,*The Cantor-Lebesgue property*, Israel J. Math.**84**(1993), no. 1-2, 179–191. MR**1244667**, 10.1007/BF02761699**[B]**J. Bourgain,*Spherical summation and uniqueness of multiple trigonometric series*, Internat. Math. Res. Notices 1996, no. 3, 93-107. CMP**96:11****[C1]**P. J. Cohen,*Topics in the theory of uniqueness of trigonometrical series,*Thesis, University of Chicago, Chicago, IL, 1958.**[C2]**Bernard Connes,*Sur les coefficients des séries trigonométriques convergentes sphériquement*, C. R. Acad. Sci. Paris Sér. A-B**283**(1976), no. 4, Aii, A159–A161 (French, with English summary). MR**0422991****[C3]**Roger Cooke,*A Cantor-Lebesgue theorem in two dimensions*, Proc. Amer. Math. Soc.**30**(1971), 547–550. MR**0282134**, 10.1090/S0002-9939-1971-0282134-X**[R]**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157****[S]**Victor L. Shapiro,*Uniqueness of multiple trigonometric series*, Ann. of Math. (2)**66**(1957), 467–480. MR**0090700**

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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

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