A CANTOR-LEBESGUE THEOREM WITH VARIABLE-COEFFICIENTS

If {phi(n)} is a lacunary sequence of integers, and if for each n, c(n )(x) and c(-n)(x) are trigonometric polynomials of degree n, then {c(n )(x)} must tend to zero for almost every x whenever {c(n)(x)e(i phi nx ) + c(-n)(-x)e(-i phi nx)} does. We conjecture that a similar result o ught to hold even when the sequence {phi(n)} has much slower growth. H owever, there is a sequence of integers {n(j)} and trigonometric polyn omials {P-j} such that {e(injx) - P-j(x)} tends to zero everywhere, ev en though the degree of P-j does not exceed n(j) - j for each j. The s equence of trigonometric polynomials {root n sin(2n) x/2} tends to zer o for almost every x, although explicit formulas are developed to show that the sequence of corresponding conjugate functions does not. Amon g trigonometric polynomials of degree n with largest Fourier coefficie nt equal to 1, the smallest one ''at'' x = 0 is 4(n)((2n)(n))(-1) sin( 2n) (x/2), while the smallest one ''near'' x = 0 is unknown.