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Weight function

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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, $\sigma$ 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://www.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