Master Dissertation

Therefore φ(x)d(x) also has quasi-asymptotics of order ω < 0 at x = 0
with respect to ρ.
Hence by (7.8) and (7.12)
〈ψ0( x
)d(x), φ(x)〉 → 0 for δ → 0 (7.14)
δ
for all φ ∈ S(R m ).
Now for fixed a1 > 1 the convergence is uniform on the compact interval
1 ≤ a ≤ a1.
We want to show the uniformity of the limit in a ′ ≥ a1 > 1.
To this end, we claim that
Claim.
χ0
n v · x
v · x
n−1
a − χ0 =
δ
δ
j=0
ψ0
j x
a
δ
We prove this by induction. It holds for n = 1 by definition. Assume it
holds for n = k − 1. Then,
k v · x
v · x
χ0 a − χ0
δ
δ
k v · x
k−1 v · x
k−1 v · x
v · x
= χ0 a − χ0 a + χ0 a − χ0
δ
δ
δ
δ
k−1 x
k−2
j x
= ψ0 a + ψ0 a .
δ
δ
j=0
Which proves the claim.
We write a ′ = a n , for a suitable n and 1 ≤ a ≤ a1. Now the problem is
translated into showing uniformity in n. By (7.13) we may choose δ0 such
that for δ < δ0
|ρ(δ)〈φ(x)d(x), ψ1(x/δ)ψ0(x/δ)〉| ≤ |〈φ(0)d0(x), ψ1(x)ψ0(x)〉| + 1 = K.
Note that δ/a j ≤ δ < δ0. Thus
From (7.5) we know that
Now since a −ω > a −ω
1
Since δ/a q < δ for all q ≥ 1
|〈φ(x)d(x), ψ1(a j x/δ)ψ0(a j x/δ)〉| ≤ K/ρ(δ/a j )
ρ(δ/a)
ρ(δ) .
for ω ≤ 0 we may assume, that for δ < δ0
ρ(δ/a) ≥ a −ω
1 ρ(δ).
ρ(δ/a j ) ≥ a −ω
1 ρ(δ/aj−1 ) ≥ a 2ω
1 ρ(δ/a j−2 ) ≥ a −jω
1
56
ρ(δ).