Orthogonal polynomials on $[0,1]$

Question

Are the orthogonal polynomials for the standard $L^2$ product on $[0,1]$ well-known? I couldn't find anything upon a quick web search.

Answer 1

The Legendre polynomials $P_n(x)$ (see this Wikipedia article) are orthogonal on $[-1,1]$. So you can just set $Q_n(x)=P_n(2x-1)$ to get an orthogonal famiy on $[0,1]$.

Answer 2

Hint: Try for this $\{f_{n}(x)=e^{2\pi i n x}: n\in \mathbb Z \}$.