Approximating linear functionals

Suppose we have a linear functional .
We want to approximate it by:

Then using Lagrange Cardinal Polynomials we find:

so we choose .

Gaussian quadrature

Suppose .
Then we can find and s.t.

is exact for .
Pick to be roots of
where is the th Orthogonal polynomials for this weight function,
and .
First, by picking we find
because has degree ,
so all are positive.

All roots of are distinct and in .

Proof

Suppose has roots .
Define .
If then is orthogonal to ,
so .
But is always positive
because they change signs at exactly the same points
so this is impossible.

Now for any polynomial
we can write it as
where and have degrees at most .
Hence

but is exact by choice of .