  We show that a bounded function $m$ on $\R$ not necessarily integrable at
infinity may still yield $L^p$-bounded convolution operators for the Jacobi
transform if the nontangential boundary values of $\omega m$ along the edges of
a certain strip in $\C$ yield Euclidean Fourier multipliers, for $\omega$
suitably defined. This partially generalizes similar results by Giulini,
Mauceri, and Meda (on rank one symmetric spaces) and Astengo (on Damek--Ricci
spaces).
