 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 349, Number 4, April 1997, Pages 16631674
S 00029947(97)016413
ONE AND TWO DIMENSIONAL
CANTORLEBESGUE TYPE THEOREMS
J. MARSHALL ASH AND GANG WANG
ABSTRACT. Let (p(n) be any function which grows more slowly than exponen
tially in n, i.e., limsupp(n)1/' < 1. There is a double trigonometric series
whose coefficients grow like (p(n), and which is everywhere convergent in the
square, restricted rectangular, and oneway iterative senses. Given any preas
signed rate, there is a one dimensional trigonometric series whose coefficients
grow at that rate, but which has an everywhere convergent partial sum sub
sequence. There is a one dimensional trigonometric series whose coefficients
grow like ~p(n), and which has the everywhere convergent partial sum subse
quence S2j. For any p > 1, there is a one dimensional trigonometric series
whose coefficients grow like (p(n P ), and which has the everywhere conver
gent partial sum subsequence S[jp]. All these examples exhibit, in a sense, the
worst possible behavior. If mj is increasing and has arbitrarily large gaps,
there is a one dimensional trigonometric series with unbounded coefficients
