Foundations of Data Science

Scale and wavelet coefficients equations
φ(x) = ∑ d−1
k=0 c kφ(2x − k)
∞∫
−∞
d−1
φ(x)φ(x − k)dx = δ(k)
∑
c j = 2
j=0
d−1 ∑
c j c j−2k = 2δ(k)
j=0
c k = 0 unless 0 ≤ k ≤ d − 1
d even
d−1 ∑
j=0
c 2j = d−1 ∑
c 2j+1
j=0
ψ(x) = d−1 ∑
∞∫
x=−∞
∞∫
x=−∞
∞∫
x=−∞
d−1
k=0
b k φ(x − k)
φ(x)ψ(x − k) = 0
ψ(x)dx = 0
ψ(x)ψ(x − k)dx = δ(k)
∑
(−1) k b i b i−2k = 2δ(k)
i=0
d−1 ∑
c j b j−2k = 0
j=0
d−1 ∑
b j = 0
j=0
b k = (−1) k c d−1−k
One designs wavelet systems so the above conditions are satisfied.
Lemma 11.2 provides a necessary but not sufficient condition on the coefficients of
the dilation equation for shifts of the scale function to be orthogonal. One should note
that the conditions of Lemma 11.2 are not true for the triangular or piecewise quadratic
solutions to
φ(x) = 1 2 φ(2x) + φ(2x − 1) + 1 φ(2x − 2)
2
and
φ(x) = 1 4 φ(2x) + 3 4 φ(2x − 1) + 3 4 φ(2x − 2) + 1 φ(2x − 3)
4
which overlap and are not orthogonal.
For φ(x) to have finite support the dilation equation can have only a finite number of
terms. This is proved in the following lemma.
Lemma 11.3 If 0 ≤ x < d is the support of φ(x), and the set of integer shifts, {φ(x −
k)|k ≥ 0}, are linearly independent, then c k = 0 unless 0 ≤ k ≤ d − 1.
Proof: If the support of φ(x) is 0 ≤ x < d, then the support of φ(2x) is 0 ≤ x < d 2 . If
φ(x) =
∞∑
k=−∞
362
c k φ(2x − k)