In this chapter we introduce a class of generalized functions specially suited for the study of Fourier series and differential equations provided with periodic boundary conditions. The concept of generalized function, as the name itself indicates, is used to generalize the notion of function and the usual calculus, and can be employed to construct scenarios adapted to the study of various problems of mathematical physics and their generalizations. In fact, the theory of generalized functions is intimately connected to the development of applied mathematics and theoretical physics during the first half of the twentieth century. Objects like the Dirac δ function were used in the formulation of quantum mechanics long before they were rigorously defined (see [65], [145] and [146]). Generally speaking, a generalized function is a certain type of linear functional defined on a space of test functions. The reason for this terminology will become clear as we proceed. At this point it is worth while to note that the properties of the generalized functions reflect the properties of the test functions on which they are defined. For example, a generalized function is as differentiable (in a generalized sense) as the corresponding test functions (in the usual sense). Distributions are special classes of generalized functions introduced by L.

This book is the outcome of several courses and seminar talks held at the Instituto de Matemática Pura e Aplicada (IMPA) over the years. It is a greatly modified version of a previous work by the authors, Equações Diferenciais Parciais, Uma Introdução, (Projeto Euclides, IMPA, 1978). It has a twofold purpose, namely to introduce the student to the basic concepts of Fourier analysis and provide illustrations of recent applications where these concepts were used to study various properties of the solutions of some important nonlinear evolution equations.

The text is divided into three parts. The first one, containing Chapters 1 to 3, deals with Fourier series and periodic distributions. Chapters 4 to 6 belong to the second part, which contains applications of Fourier series and periodic distributions to partial differential equations. Chapters 7 and 8, in the third part, are more advanced and deal with some nonperiodic problems.

Chapter 1 presents some very classical material on PDEs, such as classification into types, separation of variables and maximum principles for the heat and Laplace equations. It is by no means a comprehensive account of such topics. Rather, it's purpose is to establish the basic language used throughout the work and to provide a collection of definitions and results needed in the remainder of the book. The following two chapters deal with Fourier series and some of its applications, first in a classical setting and then in the scenario provided by p′, the space of periodic distributions.