Weight function

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".