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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 349, Number 4, April 1997, Pages 1663-1674
S 0002-9947(97)01641-3
ONE AND TWO DIMENSIONAL
CANTOR-LEBESGUE 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 one-way 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

  

Source: Ash, J. Marshall - Department of Mathematical Sciences, DePaul University

 

Collections: Mathematics