Fourier-Stieltjes coefficients and continuation of functions
Annales Academiae Scientiarum Fennicae Series A I Mathematica
1. To introduce our subject we recall two classical problems of continuation for certain functions of a complex variable. (CA) Here/is continuous in R2 and analytic outside a closed set E; E is temou' able for this problem if it is always true that/ is entire. (QC) In this problem / is a homeomorphism of the extended plane, which is K-quasiconformal outside E; E is remoaable if /is always quasiconformal in the extended plane. For both problems there is a best-possible theorem. Gehring ). Our
... purpose is to find nonremovable sets contained in FX[0, 1], not of the product type nor even approximately so. To describe these sets we denote by ,f a compact set in R2 meeting each line x:xo 4t most once, so that I is the graph of a real function whose domain is a compact set in R; since .l' is closed, that function is continuous. Theorem. (a) In each compact set E1XE2, where E1 is uncountable and E2 has positiae linear measure, there is a gtaph f non-remouable for (CA). (b) In each set E'rx[0, Ll, where El is uncountable, there is a graph l ' non-remoaable for (QC). The reason for requiring an interval on the y-axis, and not merely a set of posi tive measure in (b), can be seen from [1, p. 128]. Perhaps the correctclass of sets.Ea could be found. When .82 is an interval, the non-removability of .f can be improved in two directions. 2. Both proofs are based on a theorem of wiener (1924) about the Fourier-Stieltjes transforms of measures on R; a streamlined version of this theorem is presented in [6; p.421. We adopt the symbol e(t)=ez"it and the notation fr(u)= I e(-ut)' p(dt). Wiener's theorem is then the relation limp(2N*l)-tZ]r* p(k):p(Z), and in fact is an easy consequence of dominated convergence. When p is continuous, i,e.