We use the well-known observation that the solutions of Jacobi's differential equation can be represented via the 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 $r^d$ fast Fourier transforms (FFTs), where $r=O(\frac{\log n}{\log \log n})$ and d is the dimension, resulting in a nearly optimal algorithm to compute the multidimensional Jacobi polynomial transform.

我们使用众所周知的观察结果，即可以通过非振荡相位和振幅函数来表示Jacobi微分方程的解，从而开发了一种用于计算多维Jacobi多项式变换的快速算法。更明确地，从该观察得出，与离散雅可比变换相对应的矩阵是数值低秩矩阵与多维离散傅里叶变换（DFT）矩阵的Hadamard乘积。Hadamard产品的应用可通过以下方式进行${\mathrm{[R}}^d$ 快速傅立叶变换（FFT），其中 $\mathrm{[R}=Ø（\frac{\mathrm{日志}ñ}{\mathrm{日志}\mathrm{日志}ñ}）$和d是维数，从而产生接近最佳算法来计算所述多维雅可比多项式变换。