  This is the material for two lectures given at Ecole Polytechnique in May
2011 for the math teachers of "classes pr\'eparatoires"(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 Cand\`es, 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 Cand\`es, 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.
