A Cantor-lebesgue Theorem with Variable "coefficients"

If {qn} is a lacunary sequence of integers, and if for each n, cn(x) and c-n(x) are trigonometric polynomials of degree n, then {Cn(X)} must tend to zero for almost every x whenever {cn(x)ei?nX + c-n(-x)e-i?'nX} does. We conjecture that a similar result ought to hold even when the sequence {f On} 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 unknown.