ABSTRACT
We suppose that (φn) is a sequence of refinable functions as in Section 1.
Theorem 3.1 If P ′n(−1) = 0, then
lim n→∞∆φn∆φ̂n =
1 2 . (3.1)
Proof. We recall that limn→∞ φ˜n = G in Lp(R), 1 ≤ p ≤ ∞. From (1.6) we see that (3.1) is equivalent to
1 2 . (3.2)
We also recall that φ˜n is defined so that∫ ∞ −∞
x2φ˜n(x)dx = ∫ ∞ −∞
x2G(x)dx = 1. (3.3)
Firstly we shall show that
x2φ˜n(x)2dx = ∫ ∞ −∞
x2G(x)2dx. (3.4)
Take > 0 and choose A > 1 so that∫ |x|>A
x2G(x)dx < . (3.5)
Choose N so that for all n > N and |x| ≤ A,
|x2φ˜n(x)k − x2G(x)k| < 2A, k = 1, 2. (3.6)
Take any n > N . Then∣∣∣∣∫ A−A x2φ˜n(x)dx− ∫ A −A
x2G(x)dx ∣∣∣∣ <
and so by (3.3),∣∣∣∣∣ ∫ |x|>A
x2φ˜n(x)dx− ∫ |x|>A
x2G(x)dx
∣∣∣∣∣ < . So by (3.5), ∫
|x|>A x2φ˜n(x)dx < 2ε.
