  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.
