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I've successfully shown that Legendre polynomials are an orthonormal basis for functions. However, I'm wondering how to proof that all functions f(x) can be written in the form (Gauss-Hermite series):

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where $a$ are constants are $e$ are the Legendre polynomials.

The First StyleBender
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    Please do not repost a question after it is poorly received. Edit the original to bring up the quality. –  Oct 21 '17 at 19:36

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Firstly note that $\psi_n (x) = \frac{1}{\sqrt{2^n n!}} \frac{1}{\sqrt[4]{\pi}} e^{-\frac12 x^2}e_n(x)$, $n \in \mathbb N$ are all the eigenfunctions of the quantum harmonic oscillator: $$ - \frac12 \psi_n ''(x) +\frac12 x^2 \psi_n(x) = E_n\psi_n(x), \qquad E_n=n+\frac12. $$ We know that such functions constitute a basis for $L^2(\mathbb R)$, and your result follows for $f \in L^2(\mathbb R)$.

Martin
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  • Um... is there another way to prove this? I haven't learn about this yet. – The First StyleBender Oct 21 '17 at 20:42
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    I don't expect that to be the case. You might have a look at Stone-Weierstrass https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem (but then you got some work to do) - this is easier (in my mind) – Martin Oct 21 '17 at 20:55