论文标题
Exp $（β\ sqrt {1-z^2}）$ bernel的异叠错误（β\ sqrt
Aliasing error of the exp$(β\sqrt{1-z^2})$ kernel in the nonuniform fast Fourier transform
论文作者
论文摘要
非均匀快速傅立叶变换（NUFFT）最受欢迎的算法使用核$ ϕ $扩张在给定的非均匀点和均匀的上采样网格之间扩展（或插入式），并在频率空间中结合FFT和对角线缩放（反向尺度）。最近的Finufft库的高性能部分是由于它使用了新的“半圆的指数”内核$ ϕ（z）= e^{β\ sqrt {1-z^2}} $，对于$ z \ in [-1,1] $，零，零，否则，其fornier transform $ \ hat frons $ \ hait frons $ \ hat frons $ \ hat frons $ \ hait hat frons $均在ensanning neshnaty neflenty neflenty neflenty neflenty在家中。我们通过证明一个混蛋误差估计值将此内核放置在严格的基础上，该误差估算范围界定了1型NUFFT的误差，即确切的算术。在以上采样的网格点测得的内核宽度中，误差显示出随着通俗的kaiser-bessel-bessel内核而任意接近的指数速率降低的。这就需要使用最陡的下降，其他轮廓积分上的其他经典估计以及分阶段的SINC总和来控制$ \ hat ϕ $的尾巴上的有条件的总和。我们还在上述内核，kaiser-bessel和praly spheroidal波函数之间绘制新的连接，这些连接似乎具有最佳的指数收敛速率。
The most popular algorithm for the nonuniform fast Fourier transform (NUFFT) uses the dilation of a kernel $ϕ$ to spread (or interpolate) between given nonuniform points and a uniform upsampled grid, combined with an FFT and diagonal scaling (deconvolution) in frequency space. The high performance of the recent FINUFFT library is in part due to its use of a new "exponential of semicircle" kernel $ϕ(z)=e^{β\sqrt{1-z^2}}$, for $z\in[-1,1]$, zero otherwise, whose Fourier transform $\hatϕ$ is unknown analytically. We place this kernel on a rigorous footing by proving an aliasing error estimate which bounds the error of the one-dimensional NUFFT of types 1 and 2 in exact arithmetic. Asymptotically in the kernel width measured in upsampled grid points, the error is shown to decrease with an exponential rate arbitrarily close to that of the popular Kaiser--Bessel kernel. This requires controlling a conditionally-convergent sum over the tails of $\hatϕ$, using steepest descent, other classical estimates on contour integrals, and a phased sinc sum. We also draw new connections between the above kernel, Kaiser--Bessel, and prolate spheroidal wavefunctions of order zero, which all appear to share an optimal exponential convergence rate.
