A Characterization of Measures Whose Fourier-Stieltjes Transforms Vanish At Infinity.

Let R denote the class of complex Borel measures on the circle whose Fourier-Stieltjes coefficients (mu)(n) tend to 0 as (VBAR)n(VBAR) (--->) (INFIN). Ju. A. Sre(')ider has defined a class of subsets of , called W-sets, using the notion of asymptotic distribution. Our central result is that a measure (mu) lies in R if and only if (mu)E = 0 for all W-sets E. This establishes a claim of Sre(')ider. Rajchman conjectured that (mu) (ELEM) R if and only if (mu)E = 0 for all H-sets E and Kahane and Salem made a similar conjecture about non-normal sets. Both of these conjectures are shown to be false. Similar questions are investigated for Helson sets and weak Dirichlet sets. The characterization of R stated above extends to locally compact abelian groups. A consequence is that every measure (mu) may be split uniquely as (mu) = (mu)(,R) + (mu)(,J), where (mu)(,R) (ELEM) R, (mu)(,J) belongs to a class J, and (mu)(,R) (PERP) (mu)(,J). Any Riesz product on a compact abelian group lies purely in R or purely in J. On the circle, certain classes of infinite convolutions also have this purity property.PhDMathematicsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/159601/1/8324238.pd