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

[math]\displaystyle{ y_{n+1}= (x-c_n)y_n - d_n y_{n-1} }[/math]

for some numbers c_n and d_n, then the polynomials y_n form an orthogonal sequence for some linear functional 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(y2n) = d_n Λ(y2n–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.

See also

• Jacobi operator
• James Alexander Shohat

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, https://books.google.com/books?id=IkCJSQAACAAJ  Reprinted by Dover 2011, ISBN:978-0-486-47929-3
• Favard, J. (1935), "Sur les polynomes de Tchebicheff." (in French), C. R. Acad. Sci. Paris 200: 2052–2053, https://gallica.bnf.fr/ark:/12148/bpt6k3152t/f2052.item 
• 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 
• Hazewinkel, Michiel, ed. (2001), "Favard theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=f/f038300 
• Shohat, J. (1938), "Sur les polynômes orthogonaux généralises." (in French), C. R. Acad. Sci. Paris 207: 556–558