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Difference between revisions of "Favard theorem"

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then there is a function $\alpha(x)$ of bounded variation such that
 
then there is a function $\alpha(x)$ of bounded variation such that
  
$$\int\limits_{-\infty}^\infty P_n(x)P_m(x)d\alpha(x)=\begin{cases}0,&n\neq m,\\h_n>0,&m=n.\end{cases}$$
+
$$\int\limits_{-\infty}^\infty P_n(x)P_m(x)\,d\alpha(x)=\begin{cases}0,&n\neq m,\\h_n>0,&m=n.\end{cases}$$
  
 
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"> 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></table>
+
<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=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