ABSTRACT
This chapter begins with some facts concerning periodic functions and periodic extensions of functions. It introduces a generalization of the theory of Fourier series, namely the Sturm-Liouville theory. This theory can be used when the heat conducting material in a rod has a possibly variable thermal conductivity, density, specific heat, or source term which is proportional to the temperature. The chapter learns how to find an approximation, of the appropriate form, for any "reasonably nice" function on a finite interval. It contains statements and proofs of various convergence results for Fourier series. The square of the length of a vector in space equals the sum of the squares of its components. The chapter introduces one last definition, which is convenient in the formulation of the convergence theorem for Fourier series of piecewise C1 functions. The Sturm-Liouville theory provides a wealth of information concerning the eigenvalues and the zeros and oscillation properties of the corresponding eigenfunctions.
