Fourier-Legendre expansion of a polynomial function

Question

Why is the Fourier-Legendre expansion of a polynomial function defined on the interval (—1, 1) necessarily a finite series? I have an idea of why it might be, but I'm looking for a more "algebraic" proof.

Answer 1

For every $n=0,1,2,\dots$, let $P_n(x)=\sum\limits_{k=0}^n a_{n,k} x^k$ be a polynomial of degree $n$ (so $a_{n,n}\not=0$).

Every polynomial of degree $n$ can be written as a linear combination of $P_0,\dots, P_n$. To see this, interpret polynomials as vectors with the components being the coefficients. Then the claim follows because the $(n+1)\times (n+1)$ matrix given by $(a_{i,j})_{i,j=0,\dots,n}$ is invertible (it is lower triangular with every diagonal entry being non-zero).

This holds in particular if the $(P_n)_n$ are the Legendre polynomials.