Abstract:Let-iA be the generator of a C-0-group (U(s))(s is an element of R) on a Banach space X and omega > theta(U), the group type of U. We prove a transference principle that allows to estimate parallel to f(A)parallel to in terms of the L-p(R; X)-Fourier multiplier norm of f(. +/- i omega). If X is a Hilbert space this yields new proofs of important results of McIntosh and Boyadzhiev-de Laubenfels. If X is a UMD space, one obtains a bounded H-1(infinity)-calculus of A on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally it is proved that each generator of a cosine function on a UMD space has bounded H-infinity-calculus on sectors.