  Let $A$ be the generator of a strongly continuous cosine family $(\cos
(tA))_{t\in {\bf R}}$ on a complex Banach space $E$. The paper develops an
operational calculus for integral transforms and functions of $A$ using the
generalized harmonic analysis associated to certain hypergroups. It is shown
that characters of hypergroups which have Laplace representations give rise to
bounded operators on $E$. Examples include the Mellin transform and the
Mehler--Fock transform. The paper uses functional calculus for the cosine
family $\cos( t\sqrt {\Delta})$ which is associated with waves that travel at
unit speed. The main results include an operational calculus theorem for
Sturm--Liouville hypergroups with Laplace representation as well as analogues
to the Kunze--Stein phenomenon in the hypergroup convolution setting.
