Difference between revisions of "Favard theorem"
From Encyclopedia of Mathematics
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''on orthogonal systems'' | ''on orthogonal systems'' | ||
| − | If the following recurrence relation holds for real numbers | + | 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 | + | 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. | It was established by J. Favard [[#References|[1]]]. Sometimes this result is also linked with the name of J. Shohat. | ||
Revision as of 16:05, 30 July 2014
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) |
How to Cite This Entry:
Favard theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Favard_theorem&oldid=32576
Favard theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Favard_theorem&oldid=32576
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article