Editing Legendre polynomials (section)

=== Completeness ===

That the polynomials are complete means the following. Given any [[piecewise]] continuous function <math> f(x) </math> with finitely many discontinuities in the interval {{closed-closed|−1, 1}}, the sequence of sums
<math display="block"> f_n(x) = \sum_{\ell=0}^n a_\ell P_\ell(x)</math>
converges in the mean to <math> f(x) </math> as <math> n \to \infty </math>, provided we take
<math display="block"> a_\ell = \frac{2\ell + 1}{2} \int_{-1}^1 f(x) P_\ell(x)\,dx.</math>

This completeness property underlies all the expansions discussed in this article, and is often stated in the form
<math display="block">\sum_{\ell=0}^\infty \frac{2\ell + 1}{2} P_\ell(x)P_\ell(y) = \delta(x-y), </math>
with {{math|−1 ≤ ''x'' ≤ 1}} and {{math|−1 ≤ ''y'' ≤ 1}}.