Inner Product Spaces and Orthogonal Functions

1 Background We begin by recalling the solution of the vibrating string problem and Sturm-Liouville problems. When we solve the problem of the vibrating string using the technique of separation of variables, the differential equation involving the space variable x, and assuming constant mass density, is y (x) + ω 2 c 2 y(x) = 0, (1.1) which we can write as an eigenvalue problem y (x) + λy(x) = 0. (1.2) The solutions to Equation (1.1) are y(x) = α sin ω c x. In the vibrating string problem, the string is fixed at both ends, x = 0 and x = L, so that φ(0, t) = φ(L, t) = 0, for all t. Therefore, we must have y(0) = y(L) = 0, so that the eigenfunction solution that corresponds to the eigenvalue λ m = πm L 2 must have the form y(x) = A m sin ω m c x = A m sin πm L x , 1