Page 331 - Elements of Distribution Theory

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10.4 Gaussian Quadrature 317

The Laguerre polynomials, like the Hermite polynomials, are complete (Andrews et al.
1999, Chapter 6).

Example 10.10 (Series expansion of a density). In Example 10.9, a series expansion of
a density function based on the Hermite polynomials was considered. The same approach
may be used with the Laguerre polynomials.
Let p denote a density function on (0, ∞) and let F denote the distribution function of
the standard exponential distribution. Assume that

∞
2
p(x) exp(x) dx < ∞.
0
Under this assumption
2
p(x)
∞
dF(x) < ∞
exp(−x) 2
0
so that the function p(x)/exp(−x) has an expansion of the form
∞
p(x)
= α j L j (x)
exp(−x)
j=0
where the constants α 0 ,α 1 ,... are given by

∞
α j = L j (x)p(x)exp(−x) dx;
0
note that

∞
2
L j (x) exp(−x) dx = 1
0
and that α 0 = 1.
Hence, the function p has an expansion of the form

∞

p(x) = exp(−x) 1 + α j L j (x) .
j=1

10.4 Gaussian Quadrature

One important application of orthogonal polynomials is in the development of methods of
numerical integration. Here we give only a brief description of this area; see Section 10.6
for references to more detailed discussions.
Consider the problem of calculating the integral

b
f (x) dF(x)
a
where F is a distribution function on [a, b] and −∞ ≤ a < b ≤∞.
Let {p 0 , p 1 ,...} denote orthogonal polynomials with respect to F. Fix n and let a < x 1 <
x 2 < ··· < x n < b denote the zeros of p n . Then, by Theorem 10.5, there exist constants