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We use the well-known observation that the solutions of Jacobi's differentialequation can be represented via non-oscillatory phase and amplitude functionsto develop a fast algorithm for computing multi-dimensional Jacobi polynomialtransforms. More explicitly, it follows from this observation that the matrixcorresponding to the discrete Jacobi transform is the Hadamard product of anumerically 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 algorithmto compute the multidimensional Jacobi polynomial transform.

Fast Algorithms for the Multi-dimensional Jacobi Polynomial Transform