  We use the well-known observation that the solutions of Jacobi's differential
equation can be represented via non-oscillatory phase and amplitude functions
to develop a fast algorithm for computing multi-dimensional Jacobi polynomial
transforms. More explicitly, it follows from this observation that the matrix
corresponding to the discrete Jacobi transform is the Hadamard product of a
numerically low-rank matrix and a multi-dimensional discrete Fourier transform
(DFT) matrix. The application of the Hadamard product can be carried out via
$O(1)$ fast Fourier transforms (FFTs), resulting in a nearly optimal algorithm
to compute the multidimensional Jacobi polynomial transform.
