Operational calculus and integral transforms for groups with finite propagation speed

Let A be the generator of a strongly continuous cosine family (cos tA)) on a complex Banach space E. The paper develops an operational calculus for integral transforms and functions of A using the generalized harmonic analysis assocaited to certain hypergroups. It is shown that characters of hypergroups which have laplace representation give rise to bounded operators on E. Examples include the Mellin trasnform and the Mehler--Fock transform. The paper uses functional caclulus for the cosine family that 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