The problem of reconstructing a non-negative function from finitely many values of its Fourier transform is a problem of approximating one function by another and, as such, is analogous to the design of finite-impulse-response approximations to the Wiener filter. Using this analogy the authors obtain reconstruction methods that are computationally simpler approximations of entropy-based procedures. Their linear estimators allow for the inclusion of prior information about the oversampling rate, i.e. support information, as well as other prior knowledge of the general shape of the object. Their nonlinear methods, designed to recover spiky objects, make use of prior information about non-uniformity in the background to avoid bias in the estimation of peak locations.