Aliasing error of the exp$(\beta \sqrt{1-z^2})$ kernel in the nonuniform fast Fourier transform

The most popular algorithm for the nonuniform fast Fourier transform (NUFFT) uses the dilation of a kernel $\phi$ 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 $\phi(z)=e^{\beta \sqrt{1-z^2}}$, for $z\in[-1,1]$, zero otherwise, whose Fourier transform $\hat\phi$ 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 $\hat\phi$, 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.