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

DOI:
https://doi.org/10.1090/S0002-9947-97-01641-3

MathSciNet review:
1357390

Full-text PDF

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. M. Ash and G. V. Welland,*Convergence, uniqueness, and summability of multiple trigonometric series*, Trans. Amer. Math. Soc.**163**(1972), 401-436. MR**45:9057**.**[AFR]**J. M. Ash, C. Freiling, and D. Rinne,*Uniqueness of rectangularly convergent trigonometric series,*Ann. of Math.**137**(1993), 145-166. MR**93m:42002****[AKR]**J. M. Ash, R. P. Kaufman, and E. Rieders,*The Cantor-Lebesgue property,*Israel J. Math.**84**(1993), 179-191. MR**94m:42007****[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,**283**(1976), 159-161. MR**54:10975**.**[C3]**R. Cooke,*A Cantor-Lebesgue theorem in two dimensions,*Proc. Amer. Math. Soc.,**30**(1971), 547-550. MR**43:7847**.**[R]**W. Rudin,*Real and Complex Analysis,*3rd ed., McGraw-Hill, New York, 1987. MR**88k:00002****[S]**V. L. Shapiro,*Uniqueness of multiple trigonometric series,*Ann. of Math.**66**(1957), 467-480. MR**19:854d**.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
42A20,
42B99,
40A05,
40C99

Retrieve articles in all journals with MSC (1991): 42A20, 42B99, 40A05, 40C99

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:
https://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