Abstract: Recently, the butterfly approximation scheme and hierarchical approximations have been proposed for the efficient computation of integral transforms with oscillatory and with asymptotically smooth kernels. Combining both approaches, we propose a certain fast Fourier-Laplace transform, which in particular allows for an efficient evaluation of polynomials at nodes in the complex unit disk. We also apply this idea to develop a sparse Fourier transform, which leads to a fast algorithm for the computation of spherical mean values in arbitrary geometries.