Foundations of Data Science

the support of both sides of the equation must be the same. Since the φ(x−k) are linearly
independent the limits of the summation are actually k = 0 to d − 1 and
∑d−1
φ(x) = c k φ(2x − k).
k=0
It follows that c k = 0 unless 0 ≤ k ≤ d − 1.
The condition that the integer shifts are linearly independent is essential to the proof
and the lemma is not true without this condition.
One should also note that d−1 ∑
and k = d−1
2
i=0
c i c i−2k = 0 for k ≠ 0 implies that d is even since for d odd
∑d−1
∑d−1
c i c i−2k = c i c i−d+1 = c d−1 c 0 .
i=0
i=0
For c d−1 c 0 to be zero either c d−1 or c 0 must be zero. Since either c 0 = 0 or c d−1 = 0, there
are only d − 1 nonzero coefficients. From here on we assume that d is even. If the dilation
equation has d terms and the coefficients satisfy the linear equation ∑ d−1
k=0 c k = 2 and the
d
quadratic equations ∑ d−1
2 i=0 c ic i−2k = 2δ(k) for 1 ≤ k ≤ d−1,
then for d > 2 there are d −1
2 2
coefficients that can be used to design the wavelet system to achieve desired properties.
11.6 Derivation of the Wavelets from the Scaling Function
In a wavelet system one develops a mother wavelet as a linear combination of integer
shifts of a scaled version of the scale function φ(x). Let the mother wavelet ψ(x) be given
by ψ(x) = d−1 ∑
b k φ(2x − k). One wants integer shifts of the mother wavelet ψ(x − k) to
k=0
be orthogonal and also for integer shifts of the mother wavelet to be orthogonal to the
scaling function φ(x). These conditions place restrictions on the coefficients b k which are
the subject matter of the next two lemmas.
Lemma 11.4 (Orthogonality of ψ(x) and ψ(x − k)) Let ψ(x) = d−1 ∑
k=0
b k φ(2x − k). If ψ(x)
and ψ(x−k) are orthogonal for k ≠ 0 and ψ(x) has been normalized so that ∫ ∞
ψ(x)ψ(x−
−∞
k)dx = δ(k), then
∑d−1
(−1) k b i b i−2k = 2δ(k).
i=0
Proof: Analogous to Lemma 11.2.
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