  Nonuniform Fourier data are routinely collected in applications such as
magnetic resonance imaging, synthetic aperture radar, and synthetic imaging in
radio astronomy. To acquire a fast reconstruction that does not require an
online inverse process, the non-uniform fast Fourier transform (NFFT), also
called convolutional gridding, is frequently employed. While various
investigations have led to improvements in accuracy, efficiency, and robustness
of the NFFT, not much attention has been paid to the fundamental analysis of
the scheme, and in particular its convergence properties. This paper analyzes
the convergence of the NFFT by casting it as a Fourier frame approximation. In
so doing, we are able to design parameters for the method that satisfy
conditions for numerical convergence. Our so called frame theoretic
convolutional gridding algorithm can also be applied to detect features (such
as edges) from non-uniform Fourier samples of piecewise smooth functions.
