Page 323 - Elements of Distribution Theory

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

10.2 General Systems of Orthogonal Polynomials 309

(iv) For any constants β 0 ,β 1 ,...,β n ,
2

∞ ∞
n
ˆ 2
[ f (x) − f (x)] dF(x) ≤ β j ¯ p (x) − f (x) dF(x),
j
n
−∞ −∞ j=0
ˆ
that is, f is the best approximation to f among all polynomials of degree n, using
n
the criterion

∞
2
[g(x) − f (x)] dF(x).
−∞
ˆ
ˆ
Proof. We first show that the sequence f , f ,... converges to some function f 0 in the
2
1
sense that
∞

2
ˆ
lim ( f (x) − f 0 (x)) dF(x) = 0.
n
n→∞
−∞
Note that, for m > n,
∞
m
ˆ ˆ 2 2
[ f (x) − f (x)] dF(x) = α ;
m n j
−∞ j=n+1
hence, under the conditions of the theorem, for any > 0 there exists an N such that
∞

2
ˆ
ˆ
[ f (x) − f (x)] dF(x) ≤
n
m
−∞
for n, m > N.
The construction of the function f 0 now follows as in the proof of Theorem 6.4; hence,
only a brief sketch of the argument is given here.
There exists a subsequence n 1 , n 2 ,... such that
∞ 1

2
ˆ
[ f ˆ (x) − f (x)] dF(x) ≤ , j = 1, 2,....
n j+1 n j j
−∞ 4
For each m = 1, 2,..., define a function T m by
m

ˆ
T m (x) = | f ˆ (x) − f (x)|.
n j+1 n j
j=1
Then, for each x, either T 1 (x), T 2 (x),... has a limit or the sequence diverges to ∞. Define
T (x) = lim T m (x)
m→∞
if the limit exists; otherwise set T (x) =∞.Asin the proof of Theorem 6.4, it may be shown
that the set of x for which T (x) < ∞ has probability 1 under F; for simplicity, assume that
T (x) < ∞ for all x.It follows that
∞

ˆ
[ f ˆ (x) − f (x)]
n j+1 n j
j=1
converges absolutely and, hence, we may define a function
∞

ˆ ˆ ˆ ˆ
f 0 (x) = f (x) + [ f (x) − f (x)] = lim f (x).
n 1 n j+1 n j n j
j→∞
j=1