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 selfadjoint operators, but don't know the details ...
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closed as not a real question by David Roberts, Igor Rivin, Bill Johnson, Mark Sapir, Ryan Budney Dec 27 '11 at 20:21It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. 

