# Favard theorem

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on orthogonal systems

If the following recurrence relation holds for real numbers $\alpha_n$ and $\beta_n$:

$$P_n(x)=(x-\alpha_n)P_{n-1}(x)-\beta_nP_{n-2}(x),$$

$$P_{-1}(x)=0,\quad P_0=1,$$

then there is a function $\alpha(x)$ of bounded variation such that

$$\int\limits_{-\infty}^\infty P_n(x)P_m(x)\,d\alpha(x)=\begin{cases}0,&n\neq m,\\h_n>0,&m=n.\end{cases}$$

It was established by J. Favard [1]. Sometimes this result is also linked with the name of J. Shohat.

#### References

 [1] J. Favard, "Sur les polynomes de Tchebicheff" C.R. Acad. Sci. Paris , 200 (1935) pp. 2052–2053 [2] G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)

#### Comments

The theorem had previously been stated by Wintner (1926) and Stone (1932).

#### References

 [a1] Mourad Ismail, "Classical and Quantum Orthogonal Polynomials in One Variable", Encyclopedia of mathematics and its applications 98 , Cambridge University Press (2005) ISBN 0-521-78201-5
How to Cite This Entry:
Favard theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Favard_theorem&oldid=43634
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article