  We consider the problem of estimating a Fourier-sparse signal from noisy
samples, where the sampling is done over some interval $[0, T]$ and the
frequencies can be "off-grid". Previous methods for this problem required the
gap between frequencies to be above 1/T, the threshold required to robustly
identify individual frequencies. We show the frequency gap is not necessary to
estimate the signal as a whole: for arbitrary $k$-Fourier-sparse signals under
$\ell_2$ bounded noise, we show how to estimate the signal with a constant
factor growth of the noise and sample complexity polynomial in $k$ and
logarithmic in the bandwidth and signal-to-noise ratio.
  As a special case, we get an algorithm to interpolate degree $d$ polynomials
from noisy measurements, using $O(d)$ samples and increasing the noise by a
constant factor in $\ell_2$.
