Page 315 - Elements of Distribution Theory

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

10.2 General Systems of Orthogonal Polynomials 301

Construction of orthogonal polynomials
In many cases, orthogonal polynomials with respect to a given distribution may be easily
constructed from the moments of that distribution.

Theorem 10.2. Let
∞

n
m n = x dF(x), n = 0, 1,....
−∞
For each n = 0, 1,..., let
x x ··· 1
 n n−1 
 m n m n−1 ··· m 0 
 
p n (x) = det  m n+1 m n ··· m 1  .
 . . . 
. . ··· .
 . . . 
m 2n−1 m 2n−2 ··· m n−1
If, for some N = 0, 1, 2,...,
 
m n−1 ··· m 0
 m n ··· m 1 
α n ≡ det  . .  = 0,
. ··· .
 . . 
m 2n−2 ··· m n−1
for n = 0, 1,..., N, then {p 0 , p 1 ,..., p N } are orthogonal polynomials with respect to F.

n
Proof. Clearly, p n is a polynomial of degree n with coefficient of x given by α n = 0.
Hence, by Theorem 10.1, it suffices to show that
∞

j
p n (x)x dF(x) dx = 0, j = 0, 1,..., n − 1.
−∞
Note that, for j = 0, 1,..., n − 1,
x x ··· x
 n+ j n+ j−1 j 
 m n m n−1 ··· m 0 
j

x p n (x) = det  m n ··· m 1 
 m n+1
. . .
 
. . ··· .
 . . . 
m 2n−1 m 2n−2 ··· m n−1
j
and, since this determinant is a linear function of x n+ j , x n+ j−1 ,..., x ,
 
m n+ j m n+ j−1 ··· m j
 m n m n−1 ···
m 0 
∞  
j
p n (x)x dF(x) dx = det  m n+1 m n ··· m 1  .
. . .
 
−∞ . . .
. . ··· .
 
m 2n−1 m 2n−2 ··· m n−1
Since the first row of this matrix is identical to one of the subsequent rows, it follows that
the determinant is 0; the result follows.
In this section, the ideas will be illustrated using the Legendre polynomials; in the
following section, other families of orthogonal polynomials will be discussed.