Difference between revisions of "Favard theorem"
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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. | ||
| + | |||
| + | ====Comments==== | ||
| + | The theorem had previously been stated by Wintner (1926) and Stone (1932). | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> J. Favard, "Sur les polynômes de Tchebicheff" ''C.R. Acad. Sci. Paris'' , '''200''' (1935) pp. 2052–2053 {{ZBL|0012.06205}}</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> | ||
Latest revision as of 20:27, 15 November 2023
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 polynômes de Tchebicheff" C.R. Acad. Sci. Paris , 200 (1935) pp. 2052–2053 Zbl 0012.06205 |
| [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 |
How to Cite This Entry:
Favard theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Favard_theorem&oldid=11205
Favard theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Favard_theorem&oldid=11205
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article