View source for Favard theorem

{{TEX|done}}
''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 [[#References|[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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> J. Favard, "Sur les polynomes de Tchebicheff"  ''C.R. Acad. Sci. Paris'' , '''200'''  (1935)  pp. 2052–2053</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> 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}}</TD></TR>
</table>