 
Summary: BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 77, Number 1, January 1971
CONVERGENCE, SUMMABILITY, AND UNIQUENESS
OF MULTIPLE TRIGONOMETRIC SERIES
BY J. MARSHALL ASH1
AND GRANT V. WELLAND2
Communicated by Paul J. Cohen, August 25, 1970
1. Relationships between methods of convergences and the growth
of coefficients. It was shown by Paul J. Cohen [l] that if a multiple
trigonometric series converges regularly at almost every point of the
&torus 2nfc
=[7T, 7r]X · · · X [7T, 7r], then its coefficients an
=ani,...fnib cannot exhibit exponential growth. A particular form of
regular convergence is square convergence. Consideration of double
series of the form
00
2 * W ( 1  cosa;)Vnv
n1
shows that Cohen's seemingly gross estimates cannot be improved.
