For some partial differential equations in physics, people may separate the variables and get some eigenfunctions. And then for any solutions for that equation, people often suppose them to be a Fourier series of those eigenfunctions. But why is it right? I think it is a little too wayward ... I know that there is some relations between this procedure and the theory of compact self-adjoint operators, but don't know the details ...
closed as not a real question by David Roberts, Igor Rivin, Bill Johnson, Mark Sapir, Ryan Budney Dec 27 2011 at 20:21