TITLE: Refined Fourier analysis: procedures, error estimates and applications AUTHORS: G. Gomez Departament de Matematica Aplicada i Analisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain email: gerard@maia.ub.es J.M. Mondelo Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain email: mondelo@ma1.upc.es C. Simo Departament de Matematica Aplicada i Analisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain email: carles@maia.ub.es ABSTRACT: The purpose of this paper is to develop and analyze Fourier analysis procedures that allow an accurate determination of the frequencies and the amplitudes when the analyzed function is quasi--periodic or close to it. The procedures and theoretical results are illustrated with examples and applications.