For a LCA group G with dual group Ĝ, let D ( G ) = D ( G ^ ) D(G) = D(\hat G) denote the convex (not closed) hull of { ⟨ x , γ ⟩ : x ∈ G , γ ∈ G ^ } \{ \langle x,\gamma \rangle :x \in G,\gamma \in \hat G\} . The set D ( G ) D(G) is the natural domain for functions that operate by composition from the class, P D 1 ( G ^ ) P{D_1}(\hat G) , of Fourier-Stieltjes transforms of probability measures on G to B ( G ^ ) 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 ) D(G) . In §1, we show (1) if F operators from P D 1 ( G ) P{D_1}(G) to B ( G ) B(G) and G is compact, then K ( z ) = lim t → 1 − F ( t z ) K(z) = {\lim _{t \to {1^ - }}}F(tz) exists for all z ∈ D ( G ) z \in D(G) and K operates from P D 1 ( G ^ ) P{D_1}(\hat G) to B ( G ^ ) B(\hat G) ; (2) if F operates from P D 1 ( G ^ ) P{D_1}(\hat G) to P D ( G ^ ) = ∪ r > 0 r P D 1 ( G ^ ) PD(\hat G) = { \cup _{r > 0}}rP{D_1}(\hat G) and G is compact, then K operates from P D 1 ( G ^ ) P{D_1}(\hat G) to P D ( G ^ ) PD(\hat G) , and so also does F − K F - K ; (3) if G = D q , q ⩾ 2 G = {{\mathbf {D}}_q},q \geqslant 2 , and F operates from P D 1 ( G ^ ) P{D_1}(\hat G) to B ( G ^ ) B(\hat G) , then F = K F = K on D ( G ) ∩ { z : | z | > 1 } 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 ( G ^ ) P{D_1}(\hat G) to B ( G ^ ) 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.