In the previous chapter, we studied generalized Fourier series or eigenfunction expansions in the general framework of Sturm-Liouville theory. In the present chapter, we consider the special case of Fourier series itself. After spelling out the basic theory, we explore the special measures to handle the requirements of function representation in various situations. Finally, we extend the idea to an infinite domain to arrive at Fourier integrals.
Let us first note that the simple ODE
y″ + λy = 0,
that we have met repeatedly in previous chapters, is a Sturm-Liouville equation, with q (x) = 0 and p (x) = r (x) = 1, in the framework of Eqn. 40.5. In this chapter, we are interested in ...