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The weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w0975301.png" /> of a system of orthogonal polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w0975302.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w0975303.png" /> is a non-decreasing bounded function on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w0975304.png" /> with infinitely many points of growth, then the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w0975305.png" />, called a weight function, uniquely defines a system of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w0975306.png" />, having positive leading coefficients and satisfying the orthonormality condition.
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The weight $d\sigma(x)$ of a system of orthogonal polynomials $\{P_n(x)\}$. If $\sigma$ is a non-decreasing bounded function on an interval $[a,b]$ with infinitely many points of growth, then the measure $d\sigma(x)$, called a weight function, uniquely defines a system of polynomials $\{P_n(x)\}$, having positive leading coefficients and satisfying the orthonormality condition.
  
The distribution function, or integral weight, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w0975307.png" /> can be represented in the form
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The distribution function, or integral weight, $\sigma$ can be represented in the form
  
<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/w/w097/w097530/w0975308.png" /></td> </tr></table>
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$$\sigma=\sigma_1+\sigma_2+\sigma_3,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w0975309.png" /> is an absolutely-continuous function, called the kernel, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w09753010.png" /> is the continuous singular component and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w09753011.png" /> is the jump function. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w09753012.png" />, then one can make the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w09753013.png" /> under the integral sign; here the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w09753014.png" /> is called the differential weight of the system of polynomials.
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where $\sigma_1$ is an absolutely-continuous function, called the kernel, $\sigma_2$ is the continuous singular component and $\sigma_3$ is the jump function. If $\sigma_2\equiv\sigma_3\equiv0$, then one can make the substitution $d\sigma(x)=\sigma_1'(x)dx$ under the integral sign; here the derivative $\sigma_1'=h$ is called the differential weight of the system of polynomials.
  
Of the three components of the distribution function, only the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097530/w09753015.png" /> affects the asymptotic properties of the orthogonal polynomials.
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Of the three components of the distribution function, only the kernel $\sigma_1$ affects the asymptotic properties of the orthogonal polynomials.
  
 
For references see [[Orthogonal polynomials|Orthogonal polynomials]].
 
For references see [[Orthogonal polynomials|Orthogonal polynomials]].
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====Comments====
 
====Comments====
The term  "weight function"  is often exclusively used for what is called here  "differential weight" .
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The term  "weight function"  is often exclusively used for what is called here  "differential weight".

Latest revision as of 16:58, 3 August 2014

The weight $d\sigma(x)$ of a system of orthogonal polynomials $\{P_n(x)\}$. If $\sigma$ is a non-decreasing bounded function on an interval $[a,b]$ with infinitely many points of growth, then the measure $d\sigma(x)$, called a weight function, uniquely defines a system of polynomials $\{P_n(x)\}$, having positive leading coefficients and satisfying the orthonormality condition.

The distribution function, or integral weight, $\sigma$ can be represented in the form

$$\sigma=\sigma_1+\sigma_2+\sigma_3,$$

where $\sigma_1$ is an absolutely-continuous function, called the kernel, $\sigma_2$ is the continuous singular component and $\sigma_3$ is the jump function. If $\sigma_2\equiv\sigma_3\equiv0$, then one can make the substitution $d\sigma(x)=\sigma_1'(x)dx$ under the integral sign; here the derivative $\sigma_1'=h$ is called the differential weight of the system of polynomials.

Of the three components of the distribution function, only the kernel $\sigma_1$ affects the asymptotic properties of the orthogonal polynomials.

For references see Orthogonal polynomials.


Comments

The term "weight function" is often exclusively used for what is called here "differential weight".

How to Cite This Entry:
Weight function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weight_function&oldid=32706
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article