Orthogonal polynomials

Remarks

Condition 2 ensures that $\int p^2 \, d\mu > 0$ for every nonzero polynomial $p$, so the integral is an inner product on the polynomials. Gram–Schmidt orthogonalization of $1, x, x^2, \dots$ then gives such a sequence, and each $p_n$ is determined up to a nonzero constant factor. The normalization $p_n(x) = x^n + \cdots$ gives the monic orthogonal polynomials, and $\int p_n^2 \, d\mu = 1$ gives the orthonormal ones. Often $d\mu = w(x) \, dx$ for a weight function $w \ge 0$ on an interval.

Example: $d\mu = dx$ on $[-1, 1]$ and $p_0 = 1$, $p_1 = x$, $p_2 = \frac12(3x^2 - 1)$, the first Legendre polynomials. The integrals of $p_0 p_1 = x$ and of $p_1 p_2$ vanish because these are odd functions. And $\int_{-1}^{1} \frac12(3x^2 - 1) \, dx = \frac12 (2 - 2) = 0$.

Non-example: the monomials $1, x, x^2$ for the same measure are not orthogonal, since $\int_{-1}^{1} 1 \cdot x^2 \, dx = \frac23 \ne 0$.