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The most popular algorithm for the nonuniform fast Fourier transform (NUFFT)uses the dilation of a kernel $\phi$ to spread (or interpolate) between givennonuniform points and a uniform upsampled grid, combined with an FFT anddiagonal scaling (deconvolution) in frequency space. The high performance ofthe recent FINUFFT library is in part due to its use of a new "exponential ofsemicircle" kernel $\phi(z)=e^{\beta \sqrt{1-z^2}}$, for $z\in[-1,1]$, zerootherwise, whose Fourier transform $\hat\phi$ is unknown analytically. We placethis kernel on a rigorous footing by proving an aliasing error estimate whichbounds the error of the one-dimensional NUFFT of types 1 and 2 in exactarithmetic. Asymptotically in the kernel width measured in upsampled gridpoints, the error is shown to decrease with an exponential rate arbitrarilyclose to that of the popular Kaiser--Bessel kernel. This requires controlling aconditionally-convergent sum over the tails of $\hat\phi$, using steepestdescent, other classical estimates on contour integrals, and a phased sinc sum.We also draw new connections between the above kernel, Kaiser--Bessel, andprolate spheroidal wavefunctions of order zero, which all appear to share anoptimal exponential convergence rate.

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