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