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Uniqueness for multiple trigonometric and Walsh series with convergent rearranged square partial sums

Author(s): J. Marshall Ash; Sh. T. Tetunashvili
Journal: Proc. Amer. Math. Soc. 134 (2006), 1681-1686.
MSC (2000): Primary 42B99, 42B05, 42C20; Secondary 42C10, 42A63
Posted: December 2, 2005
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Abstract: If at each point of a set of positive Lebesgue measure every rearrangement of a multiple trigonometric series square converges to a finite value, then that series is the Fourier series of a function to which it converges uniformly. If there is at least one point at which every rearrangement of a multiple Walsh series square converges to a finite value, then that series is the Walsh-Fourier series of a function to which it converges uniformly.

References:

[AFR]
J. M. Ash, C. Freiling, and D. Rinne, Uniqueness of rectangularly convergent trigonometric series, Ann. of Math. 137 (1993), 145-166. MR 1200079 (93m:42002)

[AT]
J. M. Ash and Sh. T. Tetunashvili, New uniqueness theorems for trigonometric series, Proc. of the Amer. Math. Soc. 128 (2000), 2627-2636. MR 1657746 (2000m:42006)

[B]
J. Bourgain, Spherical summation and uniqueness of multiple trigonometric series, IMRN, no. 3 (1996), 93-107. MR 1383751 (97b:42022)

[T]
S. Tetunashvili, On some multiple function series and the solution of the uniqueness problem for Pringsheim convergence of multiple trigonometric series, Mat. Sb. 182 (1991), 1158-1176 (Russian), Math. USSR Sbornik 73 (1992), 517-534 (English). MR 1128694 (93b:42022)

[Z]
Yu. A. Zaitsev, Cantor-Lebesgue and Fatou-Denjoy-Luzin theorems for multiple trigonometric series, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 40 (1985), no. 3, 3-8, 101 (Russian), Moscow Univ. Math. Bull. 40 (1985), no. 3, 1-7 (English). MR 0802606 (87a:42032)

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Additional Information:

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

Sh. T. Tetunashvili
Affiliation: Department of Mathematics, Georgian Technical University, Kostava str. 77, 0175 Tbilisi, Republic of Georgia
Email: stetun@hotmail.com

DOI: 10.1090/S0002-9939-05-08225-0
PII: S 0002-9939(05)08225-0
Keywords: Uniqueness, multiple trigonometric series, multiple Walsh series, rearrangements
Received by editor(s): January 4, 2005
Posted: December 2, 2005
Additional Notes: This research was partially supported by NSF grant DMS 9707011 and a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University.
Communicated by: Andreas Seeger
Copyright of article: Copyright 2005, American Mathematical Society

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