Functional calculus on Venturi for Groups with Finite Propagation Speed

Let ${\cal{M}}$ be a complete Riemannian manifold with Ricci curvature bounded below and Laplace operator $\Delta$. The paper develops a functional calculus for the cosine family $\cos(t\sqrt {\Delta})$ which is associated with waves that travel at unit speed. If $f$ is holomorphic on a Venturi shaped region, and $z^kf(z)$ is bounded for some positive integer $k$, then $f({\sqrt \Delta})$ defines a bounded linear operator on $L^p({\cal{M}})$ for some $p>2$. For Jacobi hypergroups with invariant measure $m$ the generalized Fourier transform of $f\in L^1(m)$ gives $\hat f\in H^\infty (\Sigma_\omega)$ for some strip $\Sigma_\omega$. Hence one defines $\hat f(A)$ for operators $A$ in some Banach space that have a $H^\infty (\Sigma_\omega)$ functional calculus. The paper introduces an operational calculus for the Mehler--Fock transform of order zero. By transference methods, one defines $\hat f(A)$ when $\hat f$ is a $s$-Marcinkiewicz multiplier and $e^{itA}$ is a strongly continuous operator group on a $L^p$ space for $| 1/2-1/p| <1/s$.\par