Aliasing error of the exp⁡(β1−z2) kernel in the nonuniform fast Fourier transform

Abstract

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 1 − z 2 , for z ∈ [ − 1 , 1 ] , zero otherwise, whose Fourier transform ϕ ˆ 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 ϕ ˆ , 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.