The standard monograph in this area is the book Exponential fitting by Ixaru and Vanden Berghe (Kluwer, Boston - Dordrecht - London, 2004) but a fresh look on the things is necessary because many new contributions have been accumulated meantime. With no claim that our investigation is exhaustive we consider various directions of interest, try to integrate the new contributions in a natural, easy to follow way, and also detect some open problems of acute interest. Keywords: Exponential fitting, oscillatory functions, frequency-dependent coefficients.
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