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 $\varphi (z)=e^{\beta \sqrt{1-z^2}}$, for $z\in [-1,1]$, zero otherwise, whose Fourier transform $\hat{\varphi }$ 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 $\hat{\varphi }$, 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.

非均匀快速傅里叶变换（NUFFT）最受欢迎的算法是使用核lation的扩展在给定的非均匀点和均匀的上采样网格之间扩展（或内插），并在频率空间中结合FFT和对角线缩放（解卷积）。最新的FINUFFT库的高性能部分归功于它使用了新的“半圆指数”内核$\varphi （ž）={Ë}^{\beta \sqrt{\mathrm{1个}-{ž}^2}}$，对于 $ž\in [-\mathrm{1个}，\mathrm{1个}]$，否则为零，其傅立叶变换 $\hat{\varphi }$从分析上来说是未知的。通过证明一个混叠误差估计，我们将该内核置于严格的基础上，该误差估计在精确算术中限制了类型1和2的一维NUFFT的误差。在以上采样网格点测得的核宽度上渐近地显示，误差随指数速率的减小而减小，该指数速率与流行的Kaiser-Bessel核的指数速率任意接近。这就要求控制条件收敛的总和。$\hat{\varphi }$，使用最速下降法，轮廓积分的其他经典估计法以及分阶段的Sinc和。我们还绘制了上述内核，Kaiser-Bessel和零阶扁球状波函数之间的新连接，它们似乎共享最佳的指数收敛速率。