Operating functions on $B_{0}(\hat G)$ in plane regions
Sadahiro Saeki · Tohoku Mathematical Journal · 1969
Throughout this papsr, let G be any infinite compact abelian group, and G its dual.We shall respectively denote by M(G), M 0 (G), and M A (G), the measure algebra of all bounded regular measures on G, the closed ideal of those measures μ whose Fourier-Stieltjes transforms β vanish at the infinity of /\ G, and that of the measures absolutely continuous with respect to the Haar /\ /\ /\ measure of G.We shall also denote by B[G), B 0 (G), and A(G), the function /\ algebras on G consisting of the Fourier-Stieltjes transforms of the measures in M(G) y M 0 (G), and M A (G) respectively.Let us introduce a norm on B(G) by Suppose now that C is a subset of B(G), and that F(z) is a complex-valued function defined on some set E in the complex plane.We say that F(z) operates on C if