Fundamental Theorem of Gaussian Quadrature

The fundamental theorem of Gaussian quadrature states that an -point Gaussian quadrature rule integrates polynomials through polynomial degree exactly, and its abscissas are the roots of the orthogonal polynomial of polynomial degree for the same measure. Assume a finite positive measure on the real line with infinite support and finite moments of all orders.

Let be the monic polynomial of polynomial degree orthogonal to all lower-degree polynomials, with roots , ..., . The Lagrange interpolating polynomials are

(1)

with , where is the Kronecker delta. Define the weights by . Then

(2)

for every polynomial of polynomial degree at most .

To prove this, polynomial division gives

(3)

where and have polynomial degree at most . The first term has zero integral by orthogonality and vanishes at every . The Lagrange interpolating polynomial is integrated exactly by the definition of the weights. Conversely, exactness through polynomial degree forces the monic polynomial with the quadrature nodes as its roots to be orthogonal to every polynomial of polynomial degree at most , so it is .

Exactness applied to gives

(4)

No rule using real nodes can also integrate every polynomial of polynomial degree exactly. The square of its nodal polynomial vanishes at every node but has positive integral. For Gaussian nodes, the Christoffel function also gives .

For example, with on , the two-point Legendre-Gauss quadrature rule has nodes and both weights equal to 1. It is exact through polynomial degree 3, but gives for , whose integral is .