Fourier-Sparse Interpolation without a Frequency Gap

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 l2 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 l2.