Fast algorithms for the multi-dimensional Jacobi polynomial transform

We use the well-known observation that 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 matrix corresponding discrete transform is Hadamard product numerically low-rank Fourier (DFT) matrix. The application carried out rd transforms (FFTs), where r=O(log⁡nlog⁡log⁡n) d dimension, resulting in nearly optimal compute multidimensional transform.