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Why can a function satisfying same boundary conditions as functions Un(x) of a complete set be expanded as ΣCnUn(x)?

This is a question I encountered while reading Introduction to Quantum Theory by Hendrik F. Hameka (Please don't worry, I'm not trying to cheat on my homework, I am just asking out of curiosity). The book states the following:

     "It can be shown that a function f(x) that satisfies the same boundary conditions as the functions Un(x) of a complete set can be expanded as

f(x) = Σ CnUn(x)

...

If we multiply by U*m(x) and integrate, we obtain

< Um | f > = Cm "

This second part is fairly straightforward because

U*m f(x) dx = ∫ U*m Σ CnUn(x) dx

and it follows that

< Um | f > = Cn Σ < Um | Un > = Cn Σ δn,m = Cn

However, the first part confuses me because I can only justify it when Un(x) = (2π)^(-1/2) e^(inx) (The example of a complete set given in the book, actually)

In that case, it can easily be seen that the expression Σ < Un | f > Un(x) is just the Fourier Series expansion of f(x) and thus the result is valid for the particular complete set. However this does not explain why this is valid for all complete sets: "how can you prove that this conclusion is valid not only for this particular complete set, but for all complete sets?" is my question.

Thank you in advance for answering! Please feel free to post any questions of your own if you need clarification. Also, please be advised that you should not post responses on the order of "you didn't specify an interval of integration, so your question is impossible!" - if you know enough to answer the question, then you will understand why said interval is not specified, etc. Once again, thanks!

Xellers (author) 6 years ago
My suspicion is that as a Fourier Series models a function as an infinite series of periodic sine and cosine functions, if one were to use any other complete set of functions, it would model that function as a series of periodic functions that are not sine and cosine functions, but that can themselves be modeled by infinite series of sine and cosine functions. If this is the case, then this would, in fact, hold true for any complete set of functions. The question now becomes, how can we prove it?
Xellers (author)  Xellers6 years ago
Okay, so I decided to look into a little book I have called Mathematics for Quantum Mechanics by John David Jackson and it went into some more depth by (I think) using Lagrange Multipliers in order to minimize the mean square error of the original sigma expression in order to derive Bessel's Inequality and then show that the coefficient Cn must equal < Um | f > when any series of functions from a complete set is used. However, I am still a bit confused and the math is starting to make my head spin. Does anyone think they can help to explain exactly what is going on, step by step and with full mathematical rigor?