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 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 ${\displaystyle \{y_{0},y_{1},\dots \}}$ is a sequence of polynomials, where ${\displaystyle y_{n}}$ has degree ${\displaystyle n}$ and ${\displaystyle y_{0}=1}$. 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

${\displaystyle y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}}$

for some numbers ${\displaystyle c_{n}}$ and ${\displaystyle d_{n}}$, then the polynomials ${\displaystyle y_{n}}$ form an orthogonal sequence for some linear functional ${\displaystyle \Lambda }$ with ${\displaystyle \Lambda (1)=1}$; in other words ${\displaystyle \Lambda (y_{m}y_{n})=0}$ if ${\displaystyle m\neq n}$.

The linear functional ${\displaystyle \Lambda }$ is unique, and is given by ${\displaystyle \Lambda (1)=1}$, ${\displaystyle \Lambda (y_{n})=0}$ if ${\displaystyle n>0}$.

The functional ${\displaystyle \Lambda }$ satisfies ${\displaystyle \Lambda (y_{n}^{2})=d_{n}\,\Lambda (y_{n-1}^{2})}$, which implies that ${\displaystyle \Lambda }$ is positive definite if (and only if) the numbers ${\displaystyle c_{n}}$ are real and the numbers ${\displaystyle d_{n}}$ are positive.

See also

• Jacobi operator
• James Alexander Shohat
• Jean Favard

References

• Chihara, Theodore Seio (1978), An introduction to orthogonal polynomials, Mathematics and its Applications, vol. 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, vol. 26, Oxford: Oxford University Press, pp. 15–16, ISBN 0-19-853493-0, Zbl 1072.30006
• Subbotin, Yu. N. (2001) [1994], "Favard theorem", Encyclopedia of Mathematics, EMS Press
• Shohat, J. (1938), "Sur les polynômes orthogonaux généralises.", C. R. Acad. Sci. Paris (in French), 207: 556–558, Zbl 0019.40503