CHARACTERIZATIONS OF CLASSES OF I_0SETS IN DISCRETE ABELIAN GROUPS

摘要

A subset E of a discrete abelian group is an isEI0se t hife rvesrtyr ibctoiuon doefd hfuenction on Fourier transform of a discrete measure. Special examples of ssFRI0etes tianscdl u ((drZsea h(lhr)eaol)rtI for Fatou-Zygmund interpolation sets): E is called (real) Rh [or (real) FZIo] if every bounded (real) Hermitian function on E can be interpolated by the transform of a discrete, real [respectively, nonnegative] measure. The two pairs of classes, (real) RIo and (real) FZI0 , are shown to be identical for sets not containing the identity; the class of real Rh sets is strictly smaller than the class of I0 sets, and the class of FZIo sets is strictly smaller than the real FZI0 sets. That completes the problem of determining which of these classes are different and which are the same. Topological characterizations of these classes of sets are given, as are some union results.