 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 125, Number 1, January 1997, Pages 219228
S 00029939(97)035685
A CANTORLEBESGUE THEOREM
WITH VARIABLE "COEFFICIENTS"
J. MARSHALL ASH, GANG WANG, AND DAVID WEINBERG
(Communicated by Christopher D. Sogge)
ABSTRACT. If {qn} is a lacunary sequence of integers, and if for each n, cn(x)
and cn(x) are trigonometric polynomials of degree n, then {Cn(X)} must tend
to zero for almost every x whenever {cn(x)ei?nX + cn(x)ei?'nX} does.
We conjecture that a similar result ought to hold even when the sequence
{fOn} has much slower growth. However, there is a sequence of integers {nj }
and trigonometric polynomials {Pj} such that feinj x  Pj (x)} tends to zero
everywhere, even though the degree of Pj does not exceed nj  j for each
j. The sequence of trigonometric polynomials {V sin2n x2} tends to zero
for almost every x, although explicit formulas are developed to show that the
sequence of corresponding conjugate functions does not. Among trigonometric
polynomials of degree n with largest Fourier coefficient equal to 1, the smallest
one "at" x = 0 is 4n 2n  sin2n (x), while the smallest one "near" x = 0 is
