Page 325 - Elements of Distribution Theory

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

10.2 General Systems of Orthogonal Polynomials 311

To show parts (ii) and (iii), note that

∞ ∞ ∞
ˆ 2 2 ˆ 2
( f (x) − f (x)) dF(x) = f (x) dF(x) − f (x) dF(x)
n
n
−∞ −∞ −∞
∞
n
2
2
= f (x) dF(x) − α .
j
−∞ j=0
This proves that part (ii) and part (iii) now follows from (10.8).
Finally, consider part (iv). Note that
2
n n n

∞
∞

2
2
β j ¯ p (x) − f (x) dF(x) = f (x) dF(x) − 2 β j α j + β .
j
j
−∞ j=0 −∞ j=0 j=0
Hence,
2

n
∞
∞
2
ˆ
β j ¯ p (x) − f (x) dF(x) − [ f (x) − f (x)] dF(x)
n
j
−∞ j=0 −∞
n n n

2

2
= β − 2 β j α j + α
j j
j=0 j=0 j=0
n

2
= (α j − β j ) ,
j=0
proving the result.
Hence, according to Theorem 10.6, if {p 0 , p 1 ,...} is complete and

∞
2
f (x) dF(x) < ∞,
−∞
the function f may be written
∞

p j (x)
f (x) =
∞ 2
α j
j=0 −∞ p j (x) dF(x)
for constants α 0 ,α 1 ,... given by
∞

f (x)p n (x) dF(x)
−∞
α n = .
∞ 2
p n (x) dF(x)
−∞
In interpreting the infinite series in this expression, it is important to keep in mind that it
means that
2

n
∞

p j (x)
lim f (x) − dF(x) = 0.
∞ 2
α j
n→∞ p j (x) dF(x)
−∞ j=0 −∞
It is not necessarily true that for a given value of x the numerical series
n

p j (x)
∞ 2
α j
j=0 −∞ p j (x) dF(x)
converges to f (x)as n →∞.