Quadrature rule

A quadrature rule approximates a weighted integral by a finite weighted sum of function values.

On a compact interval, positive quadrature rule weights whose sum is the interval length define uniformly bounded functionals on continuous functions. If the rules eventually integrate every fixed polynomial exactly, the Weierstrass approximation theorem proves convergence for every continuous function.

An $(n+1)$-node quadrature rule already exact through degree $n$ is exact through degree $n+1+k$ exactly when its nodal polynomialis orthogonal to every polynomial of degree at most $k$.

No $(n+1)$-node quadrature rule for a positive interval weight can be exact through degree $2n+2$, because it evaluates the nonnegative nodal square $Q_{n+1}^2$ as zero although its integral is positive.

If an $(n+1)$-node quadrature rule for a positive interval weight is exact through degree $2n$, every weight is positive. Indeed, applying the rule to the square of the corresponding degree-$n$ Lagrange cardinal polynomial isolates that weight.

If each $(n+1)$-node quadrature rule is exact through degree $n$ and has positive weights, then it converges on every continuous function. The Weierstrass approximation theorem reduces the error to the uniform polynomial-approximation error, while exactness on constants controls the sum of the weights.

A quadrature rule is exact through degree $d$ precisely when its node weights reproduce the first $d+1$ moments of the integration measure.