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From finite complex spectral data one can construct a continuous object with a given support that is consistent with the data. Given Fourier magnitude data only, one can choose the phases arbitrarily in the above construction. The energy in the extrapolated spectrum is phase dependent and provides a cost function to be used in phase retrieval. The minimization process is performed iteratively, using an algorithm that can be viewed as a combination of Gerchberg–Papoulis and Fienup error reduction.