Abstract
Let
be a locally compact group and
be a compact group of automorphisms of
.
We consider the functional equation
where
is a continuous
function,
is a weakly
continuous function and (
) is a
Hilbert space. This equation is a generalization of Gajda's
functional equation of d'Alembert type. If
is a
solution of this equation, then the functions
and
are
-invariant and
is
-positive definite, i.e. the kernel
is positive definite. This kernel is
the reproducing kernel of a Hilbert space of functions on
,
and this implies several properties for
. If
is of finite dimensional, we show that the general
solution of this equation is of the form
where
is an operator
valued
-spherical function, with
(
is
the adjoint operator of
) and
. As an
application Chojnacki's and Stetkær's results on
operator-valued spherical functions are used to give explicit
solution formulas of this equation, in terms of strongly
continuous unitary representations of