Favard's theorem

In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable 3-term recurrence relation is a sequence of orthogonal polynomials. The theorem was introduced in the theory of orthogonal polynomials by Favard (1935) and Shohat (1938), 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 y₀ = 1, y₁, ... is a sequence of polynomials where y_n has degree n. If this is a sequence of orthogonal polynomials for some positive weight function then it satisfies a 3-term recurrence relation. Favard's theorem is roughly a converse of this, and states that if these polynomials satisfy a 3-term recurrence relation of the form

y_{n+1}= (x-c_n)y_n - d_n y_{n-1}

for some numbers c_n and d_n, then the polynomials y_n form an orthogonal sequence for some linear function Λ with Λ(1)=1; in other words Λ(y_my_n) = 0 if m ≠ n.

The linear functional Λ is unique, and is given by Λ(1) = 1, Λ(y_n) = 0 if n > 0.

The functional Λ satisfies Λ(y2
n) = d_n Λ(y2
n–1), which implies that Λ is positive definite if (and only if) the numbers c_n are real and the numbers d_n are positive.

References

Chihara, Theodore Seio (1978), An introduction to orthogonal polynomials, Mathematics and its Applications 13, New York: Gordon and Breach Science Publishers, ISBN 978-0-677-04150-6, MR 0481884 Reprinted by Dover 2011, ISBN 978-0-486-47929-3
Favard, J. (1935), "Sur les polynomes de Tchebicheff.", C. R. Acad. Sci., Paris (in French) 200: 2052–2053, JFM 61.0288.01
Rahman, Q. I.; Schmeisser, G. (2002), Analytic theory of polynomials, London Mathematical Society Monographs. New Series 26, Oxford: Oxford University Press, pp. 15–16, ISBN 0-19-853493-0, Zbl 1072.30006
Subbotin, Yu. N. (2001), "Favard Theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Shohat, J. (1938), "Sur les polynômes orthogonaux généralises.", C. R. Acad. Sci., Paris (in French) 207: 556–558, Zbl 0019.40503