This is the material for two lectures given at Ecole Polytechnique in May 2011 for the math teachers of "classes préparatoires"(parallel to the undergraduate classes in universities). The introduction is a personal overview on Fourier analysis, its history, the terms in use and a few references. The first part is also an overview but on a restricted subject : the link between statistics and Fourier, from Legendre and the role of l^2 to Donoho and the role of l^1. The third part starts from a theorem of Candès, Romberg and Tao on compressive sampling and applies the basic idea to a quite different question : the reconstruction of a function whose unknown spectrum has large gaps from Its restriction to an interval. The description and extention of the method of Candès, Romberg and Tao in Appendix 2 was developed later in a note aux Comptes rendus and an article submitted to Annales de l'Institut Fourier.
