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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038300/f0383001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038300/f0383002.png" />:
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If the following recurrence relation holds for real numbers $\alpha_n$ and $\beta_n$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038300/f0383003.png" /></td> </tr></table>
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$$P_n(x)=(x-\alpha_n)P_{n-1}(x)-\beta_nP_{n-2}(x),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038300/f0383004.png" /></td> </tr></table>
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$$P_{-1}(x)=0,\quad P_0=1,$$
  
then there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038300/f0383005.png" /> of bounded variation such that
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then there is a function $\alpha(x)$ of bounded variation such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038300/f0383006.png" /></td> </tr></table>
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$$\int\limits_{-\infty}^\infty P_n(x)P_m(x)d\alpha(x)=\begin{cases}0,&n\neq m,\\h_n>0,&m=n.\end{cases}$$
  
 
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=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