Soft extrapolation of bandlimited functions

Soft extrapolation refers to the problem of recovering a function from its samples multiplied by a fast-decaying window — in this note a narrow Gaussian. The question is akin to deconvolution, but leverages smoothness of the function in order to achieve stable recovery over an interval potentially larger than the essential support of the window. In case the function is bandlimited, we provide an error bound for extrapolation by a least-squares polynomial fit of a well-chosen degree: it is (morally) proportional to a fractional power of the perturbation level, which goes from 1 near the available samples, to 0 when the extrapolation distance reaches the characteristic smoothness length scale of the function. This bound is minimax in the sense that no algorithm can yield a meaningfully lower error over the same smoothness class. The result in this note can be put in the context of blind superresolution, where it corresponds to the limit of a single spike corrupted by a compactly-supported blur.