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.