What are the necessary and sufficient condition for $f(x)$ to be expandable into Legendre polynomials?

Question

In physics, we often expand functions $f(x)$ of a real variable $x$ which is piecewise continuous in the interval $(-1,1)$ along with its derivatives, in terms of Legendre polynomials $\{P_n(x)\}$ as $$f(x)=\sum\limits_{n=0}^{\infty}c_nP_n(x)$$ where $c_n$ can be obtained using $$c_n=\frac{2n+1}{2}\int\limits_{-1}^{+1}f(x)P_n(x)dx.$$

Is it always possible to do such an expansion for any $f(x)$? I'm looking for the necessary and sufficient conditions.

The French Wikipedia article on Legendre’s polynomial provides elements on Legendre expansion of functions. It provides more elements on that topic than the English version. — mathcounterexamples.net, Jul 01 '18 at 14:34

score 1 · Answer 1 · answered Jul 03 '18 at 04:35

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If you mean that the series converges in the $L^2$ sense (common in physics), then it follows from orthogonality and completeness. This Answer derives the completeness of the polynomials from the completeness of the trigonometric functions. Proof of the latter requires some functional analysis. This Answer outlines a proof ("glossing over technical details").

answered Jul 03 '18 at 04:35

Keith McClary

1,410

Yes, every $L^2(-1,1)$ function can be expanded in terms of Legendre polynomials. – Gonzalo Benavides Jul 03 '18 at 04:53

What are the necessary and sufficient condition for $f(x)$ to be expandable into Legendre polynomials?

1 Answers1