Foundations of Data Science

The last step requires that d be even which we have assumed for all scale functions.
For the case where the wavelet is ψ(2 j − l), first express φ(x) as a linear combination
of φ(2 j−1 x − n). Now for each these terms
∫ ∞
−∞
To see this, substitute y = 2 j−1 x. Then
∫ ∞
−∞
φ(2 j−1 x − m)ψ(2 j x − k)dx = 0
φ(2 j x − m)ψ(2 j x − k)dx = 1
2 j−1 ∫ ∞
which by the previous argument is zero.
−∞
φ(y − m)ψ(2y − k)dy
The next lemma gives conditions on the coefficients b k
wavelets to be orthogonal.
that are sufficient for the
Lemma 11.8 If b k = (−1) k c d−1−k , then
∫ ∞
−∞
1
2 j ψ j(2 j x − k) 1 2 k ψ l(2 l x − m)dx = δ(j − l)δ(k − m).
Proof: The first level wavelets are orthogonal.
∫ ∞
−∞
ψ(x)ψ(x − k)dx =
Substituting j for d − 1 − i yields
∫ ∞
∑d−1
=
∑d−1
∑d−1
b i φ(2x − i) b j φ(2x − 2k − j)dx
−∞ i=0
i=0
b i
d−1
j=0
∑
∫ ∞
b j φ(2x − i)φ(2x − 2k − j)dx
j=0
−∞
∑d−1
∑d−1
= b i b j δ(i − 2k − j)
i=0
∑d−1
=
i=0
j=0
bib i−2k
∑d−1
= (−1) i c d−1−i (−1) i−2k c d−1−i+2k
i=0
∑d−1
= (−1) 2i−2k c d−1−i c d−1−i+2k
i=0
∑d−1
c j c j+2k = 2δ(k)
j=0
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