Master Dissertation

Thus
n−1
|
j=0
〈φ(x)d(x), ψ1(a j x/δ)ψ0(a j n−1
x/δ)〉| ≤
But ρ(δ) → ∞ Hence the convergence is uniform.
We now prove the theorem.
K/ρ(δ/a j )
j=0
≤ Kρ(δ) −1
∞
j=0
a jω
1
= Kρ(δ) −1 (1 − a ω 1 ) −1 .
Proof of Theorem 7.7. Another way to state the theorem is that the
limit in (7.11) exists for all sequences δn → 0. This is done by showing
〈χ0( v·x)d(x),
φ(x)〉 is a Cauchy sequence. Consider
δn
v · x v · x
v · x v · x
χ0( ) − χ0( ) = χ0(an,m ) − χ0( ), (7.15)
δm
δn
where an,m = δn/δm. Assuming δm < δn, we know from Lemma 7.8 that
for all φ and for all ɛ > 0 there exists some δ0 > 0 such that for all δ < δ0
|〈(χ0(an,m
v · x
δ
δn
δn
v · x
) − χ0( ))d(x), φ(x)〉| < ɛ
δ
Hence given φ and ɛ > 0 we may choose δ0 and N such that δn < δ0 for all
n, m ≥ N
v · x v · x
|〈(χ0( ) − χ0( ))d(x), φ(x)〉| < ɛ.
δm
δn
Hence 〈χ0( v·x)d(x),
φ(x)〉 is a Cauchy sequence.
δn
Now suppose δn and δn ′ are two sequences converging towards 0. Then as
above, with an,n ′ = δn ′/δn in (7.15),
v · x
v · x
〈χ0( )d(x), φ(x)〉 − 〈χ0( )d(x), φ(x)〉 → 0
δn
as n → ∞. Hence the limit exists.
Now we may define
Definition 7.9.
δn ′
v · x
r(x) = lim χ0( )d(x) = θ(v · x)d(x).
δ→0 δ
Similarly there exists an advanced distribution.
57