I'm now studying the application of legendre polynomials to numerical integration in the so called gaussian quadrature. There one exploits the fact that an orthogonal polynomial of degree n is orthogonal to all other polynomials of degree less than n with respect to some weight function. For legendre polynomials that must mean that(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int_{-1}^{1} L_n(x) P_m(x) dx = 0[/tex]

for all P(x) where m is less than n. How does one prove that the legendre polynomials are in the set of such orthogonal polynomials? It's okay that they are orthogonal among themselves, but I wonder how to show that they are orthogonal to everyone else with lower degree?

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# Proof that the legendre polynomials are orthogonal polynomials

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