  Soft extrapolation refers to the problem of recovering a function from its
samples, multiplied by a fast-decaying window and perturbed by an additive
noise, over an interval which is potentially larger than the essential support
of the window. A core theoretical question is to provide bounds on the possible
amount of extrapolation, depending on the sample perturbation level and the
function prior. In this paper we consider soft extrapolation of entire
functions of finite order and type (containing the class of bandlimited
functions as a special case), multiplied by a super-exponentially decaying
window (such as a Gaussian). We consider a weighted least-squares polynomial
approximation with judiciously chosen number of terms and a number of samples
which scales linearly with the degree of approximation. It is shown that this
simple procedure provides stable recovery with an extrapolation factor which
scales logarithmically with the perturbation level and is inversely
proportional to the characteristic lengthscale of the function. The pointwise
extrapolation error exhibits a H\"{o}lder-type continuity with an exponent
derived from weighted potential theory, which changes from 1 near the available
samples, to 0 when the extrapolation distance reaches the characteristic
smoothness length scale of the function. The algorithm is asymptotically
minimax, in the sense that there is essentially no better algorithm yielding
meaningfully lower error over the same smoothness class. When viewed in the
dual domain, the above problem corresponds to (stable) simultaneous
de-convolution and super-resolution for objects of small space/time extent. Our
results then show that the amount of achievable super-resolution is inversely
proportional to the object size, and therefore can be significant for small
objects.
