The weight
of a system of
orthogonal polynomials
.
If
is a non-decreasing bounded function on an interval
with infinitely many points of growth, then the measure
,
called a
weight function,
uniquely defines a system of polynomials
,
having positive leading coefficients and satisfying the orthonormality condition.
The
distribution function,
or
integral weight,
can be represented in the form
where
is an absolutely-continuous function, called the
kernel,
is the continuous
singular component
and
is the
jump function.
If
,
then one can make the substitution
under the integral sign; here the derivative
is called the
differential weight
of the system of polynomials.
Of the three components of the distribution function, only the kernel
affects the asymptotic properties of the orthogonal polynomials.
For references see
Orthogonal polynomials.