 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 134, Number 6, Pages 16811686
S 00029939(05)082250
Article electronically published on December 2, 2005
UNIQUENESS FOR MULTIPLE TRIGONOMETRIC
AND WALSH SERIES WITH CONVERGENT
REARRANGED SQUARE PARTIAL SUMS
J. MARSHALL ASH AND SH. T. TETUNASHVILI
(Communicated by Andreas Seeger)
Abstract. If at each point of a set of positive Lebesgue measure every re
arrangement 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
WalshFourier series of a function to which it converges uniformly.
1. Introduction
The basic uniqueness results for onedimensional trigonometric series have been
successfully generalized to rectangularly convergent multiple trigonometric series
and also to spherically convergent multiple trigonometric series [AFR, B, T]. For
