Functional calculus and positive-definite functions

Abstract:

For a LCA group G with dual group Ĝ, let $D(G) = D(\hat G)$ denote the convex (not closed) hull of $\{ \langle x,\gamma \rangle :x \in G,\gamma \in \hat G\}$. The set $D(G)$ is the natural domain for functions that operate by composition from the class, $P{D_1}(\hat G)$, of Fourier-Stieltjes transforms of probability measures on G to $B(\hat G)$, the class of all Fourier-Stieltjes transforms on Ĝ. Little is known about the behavior of F on the boundary of $D(G)$. In §1, we show (1) if F operators from $P{D_1}(G)$ to $B(G)$ and G is compact, then $K(z) = {\lim _{t \to {1^ - }}}F(tz)$ exists for all $z \in D(G)$ and K operates from $P{D_1}(\hat G)$ to $B(\hat G)$; (2) if F operates from $P{D_1}(\hat G)$ to $PD(\hat G) = { \cup _{r > 0}}rP{D_1}(\hat G)$ and G is compact, then K operates from $P{D_1}(\hat G)$ to $PD(\hat G)$, and so also does $F - K$; (3) if $G = {{\mathbf {D}}_q},q \geqslant 2$, and F operates from $P{D_1}(\hat G)$ to $B(\hat G)$, then $F = K$ on $D(G) \cap \{ z:|z| < 1\}$. This third result is shown to be sharp for compact groups of bounded order. In §2, an example is given that fills a gap in the theory of functions operating from $P{D_1}(\hat G)$ to $B(\hat G)$. In §3 we show that most Riesz products and all continuous measures on K-sets have a property that is very useful in proving symbolic calculus theorems. Applications of this are indicated. Some open questions are given in §4.