Favard theorem
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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://encyclopediaofmath.org/index.php?title=Favard_theorem&oldid=43634
Favard theorem. Encyclopedia of Mathematics. URL: http://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