Page 326 - Elements of Distribution Theory

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312 Orthogonal Polynomials

Example 10.7 (Series expansion for a density). Consider a absolutely continuous distri-
bution on [−1, 1]. Suppose the moments of the distribution are available, but the form of
the density is not. We may approximate the density function by a lower-order polynomial
by using an expansion in terms of the Legendre polynomials.
Consider a quadratic approximation of the form

2
a + bx + cx ;
in terms of the Legendre polynomials, this approximation is

1 2
a + c P 0 (x) + bP 1 (x) + cP 2 (x).
3 3
Using the approach of Theorem 10.6, the approximation to the density based on P 0 , P 1 , and
P 2 is of the form
3 2 2 1

P 0 (x) + µP 1 (x) + (µ + σ ) − P 2 (x)
2 2
where µ and σ are the mean and standard deviation, respectively, of the distribution.
Hence, an approximation to the density is given by
5 1 2 2 2 2 3
2
− (µ + σ ) + µx + µ + σ − x .
4 3 4
Although this function must integrate to 1, it is not guaranteed to be nonnegative. If µ = 0,
it is straightforward to show that it is nonnegative.

10.3 Classical Orthogonal Polynomials

Although orthogonal polynomials may be constructed for any distribution for which all the
moments exist, there are only a few distributions that are often used in this context. One
is the uniform distribution on (−1, 1), leading to the Legendre polynomials discussed in
the previous section. Others include the standard normal distribution, which leads to the
Hermite polynomials, and the standard exponential distribution, which leads to the Laguerre
polynomials.In this section, we consider the properties of the Hermite and Laguerre
polynomials.

Hermite polynomials
Orthogonal polynomials with respect to the standard normal distribution function are
known as the Hermite polynomials. These polynomials are often normalized by taking the
n
coefficient of x in H n (x)tobe1 and that is the standardization that we will use here.
The following result shows that the Hermite polynomials may be generated using the
procedure described in Theorem 10.3 using the function g(x) = 1. That is, we may take
n
(−1) d n
H n (x) = φ(x), n = 0, 1,... (10.9)
φ(x) dx n
where
1 1 2
φ(x) = √ exp − x , −∞ < x < ∞
(2π) 2
denotes the standard normal density function.