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CaptainBlack · #5 November 11th, 2007, 12:37 AM

Quote:

Originally Posted by ThePerfectHacker

2)Let $f(x),g(x)$ be continous on $[0,1]$ so that $\int_0^1 x^n f(x) dx = \int_0^1 x^n g(x) dx$ for all $n\geq 0$ . Prove that $f(x) = g(x)$ . (This is a classic).

Let $P_n(x);\ n \in \mathbb(Z)_+$ be an orthonormal basis of polynomials for $L^2_{[0,1]}$ .

Then by the conditions specified in the problem:

$\langle P_n, f-g \rangle =0;\ \forall n \in \mathbb{Z}_+$

which implies that $f(x)-g(x)=0\ a.e. \in [0,1]$ which as f-g is continuous implies $f(x)-g(x)=0\ x\in [0,1]$

RonL