This first volume, a three-part introduction to Fourier analysis, is intended for students with a beginning knowledge of mathematical analysis. The first part concersn notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression.