Fast Algorithms for the Multi-dimensional Jacobi Polynomial Transform

2019/01/22 by James Bremer, Bremer, James, Qiyuan Pang +3
Computer Science · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Orbital Angular Momentum in Optics

paper · pdf · doi:10.48550/arxiv.1901.07275

openalex publication_date 2019/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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