Page 313 - Elements of Distribution Theory

P1: JZP
052184472Xc10 CUNY148/Severini May 24, 2005 2:50

10

Orthogonal Polynomials

10.1 Introduction
Let F denote a distribution function on the real line such that

∞
r
|x| dF(x) < ∞
−∞
for all r = 0, 1,.... A set of functions { f 0 , f 1 ,...} is said to be orthogonal with respect to
F if

∞
f j (x) f k (x) dF(x) = 0 for j = k.
−∞
Suppose that, for each n = 0, 1,..., f n is a polynomial of degree n which we will denote
n
by p n ; assume that the coefficient of x in p n (x)is nonzero. Then {p 0 , p 1 ,...} are said to
be orthogonal polynomials with respect to F.
Orthogonal polynomials are useful in a number of different contexts in distribution
theory. For instance, they may be used to approximate functions or they may be used in
the exact or approximate calculation of certain integrals; they also play a central role in
asymptotic expansions for distribution functions, as will be discussed in Chapter 14. In this
chapter, we give the basic properties of orthgonal polynomials with respect to a distribution
function, along with some applications of these ideas.

10.2 General Systems of Orthogonal Polynomials

Let {p 0 , p 1 ,...} denote orthogonal polynomials with respect to a distribution function F.
Then any finite subset of {p 0 , p 1 ,...} is linearly independent. A formal statement of this
result is given in the following lemma; the proof is left as an exercise.

Lemma 10.1. Let {p 0 , p 1 ,...} denote orthogonal polynomials with respect to a distri-
bution function F. Then, for any integers n 1 < n 2 < ··· < n m and any real numbers
α 1 ,α 2 ,...,α m ,

(x) = 0 a.e. (F)
α 1 p n 1 (x) +· · · + α m p n m
if and only if α 1 =· · · = α m = 0.

299