Aliasing error of the exp(β√1-z2) kernel in the nonuniform fast Fourier transform

The most popular algorithm for the nonuniform fast Fourier transform (NUFFT) uses the dilation of a kernel 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 phi(z) = e(beta root 1-z2), for z is an element of [-1, 1], zero otherwise, whose Fourier transform (phi) over cap 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 (phi) over cap, 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. (c) 2020 Elsevier Inc. All rights reserved.