Discrete sampling inner product on polynomial spaces ($⟨f,g⟩={\sum }_{j=0}^n{w}_{j}f(t_j)g(t_j)$)

Let $P_n(\mathbb{R})$ be the vector space of real polynomials of degree of a polynomial at most $n$. Choose $n+1$ distinct real sample points $t_0,\dots ,t_n$ and positive real weights $w_0,\dots ,w_n$. Thenis an inner product. Symmetry and bilinearity follow from linearity of polynomial evaluation. Its value at $(f,f)$ is nonnegative, and equality forces all $f(t_j)=0$. By the root bound for a polynomial, a nonzero polynomial of degree of a polynomial at most $n$ cannot vanish at $n+1$ distinct points. Thus the form has positive definiteness.

Equivalently, the polynomial evaluation map on a finite point set identifies $P_n(\mathbb{R})$ with ${\mathbb{R}}^{n+1}$, equipped with a weighted Euclidean inner product. Too few sample points destroy positive definiteness: if the distinct points number at most $n$, the nonzero polynomial ${\prod }_j(x-t_j)$ lies in $P_n(\mathbb{R})$ and has zero sampled squared norm.

For three equally weighted samples at $-1,0,1$, an orthonormal basis isThis construction gives a finite sampling version of orthogonal polynomials and permits exact Gram-Schmidt orthogonalization using only finitely many evaluations.