View source for Weight function

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.

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

<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>

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.

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.

For references see [[Orthogonal polynomials|Orthogonal polynomials]].



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