Proof that Legendre Polynomials are Complete

Question

Can somebody either point me to, or show me a proof, that the Legendre polynomials, or any set of eigenfunctions, are complete?

Answer 1

I think that B_Scheiner's answer could be condensed to the statement that it depends on it being of Sturm–Liouville type.

Once you have a linear inner product and resolution of the identity, then $$\begin{align}f(x) &= \int\!dx'\, \delta(x-x') f(x')\\ & = \int\!dx'\, \left[ \sum_{n} u_n(x) u_n(x') w(x) \right] f(x')\\ & = \sum_n u_n(x) \left[\int\!dx'\, u_n(x') w(x) f(x') \right]\\ & = \sum_n u_n(x) f_n \end{align}$$

Thus all you really need to do is prove that the set of all Legendre polynomials satisfies $\sum_n p_n(x) p_n(y) = \delta(x-y)$ over the domain.

This properly belongs in math.se though, but I suppose it's something every physicist should know.

Answer 2

Let $M_n = \int_a^b |f(x)-\sum_i a_i f_i(x)|^2 dx$ where $f_i$ is an orthonormal set of functions (such as the legendre polynomials). The set of $ f_i$ is complete if there is a set of coefficients $\{a_i\}$ such that $\lim_{n -> \infty} M_n=0$. If you can show that you can approximate a function on a closed interval in a way such that $M_n$ goes to zero as n goes to infinity then you are golden ( maybe look into the wierstrass approximation theorem).

I should mention that the legendre polynomials are part of what are termed Sturm -Louiville problems, and your question could be generalized to a much larger set of polynomials.