Favard's theorem

In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable three-term recurrence relation is a sequence of orthogonal polynomials. The theorem was introduced in the theory of orthogonal polynomials by and, though essentially the same theorem was used by Stieltjes in the theory of continued fractions many years before Favard's paper, and was rediscovered several times by other authors before Favard's work.

Statement

Suppose that is a sequence of polynomials, where has degree and. If this is a sequence of orthogonal polynomials for some positive weight function then it satisfies a three-term recurrence relation. Favard's theorem is roughly a converse of this, and states that if these polynomials satisfy a three-term recurrence relation of the form
for some numbers and,
then the polynomials form an orthogonal sequence for some linear functional with ; in other words if.
The linear functional is unique, and is given by, if.
The functional satisfies, which implies that is positive definite if the numbers are real and the numbers are positive.