Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal I in
L
1
(
G
/
/
K
)
{L^1}(G//K)
, the space of radial integrable functions on
G
=
S
U
(
1
,
1
)
G = SU(1,1)
, so that
I
=
L
1
(
G
/
/
K
)
I = {L^1}(G//K)
or
I
=
L
0
1
(
G
/
/
K
)
I = L_0^1(G//K)
—the ideal of
L
1
(
G
/
/
K
)
{L^1}(G//K)
functions whose integral is zero. This is then used to prove a generalization of Furstenberg’s theorem which characterizes harmonic functions on the unit disk by a mean value property and a "two circles" Morera type theorem (earlier announced by Agranovskiĭ).