Favard theorem

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.

Comments

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

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)
[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