Foundations of Data Science

Lemma 11.5 (Orthogonality of φ(x) and ψ(x − k)) Let φ(x) = d−1 ∑
c k φ(2x − k) and
ψ(x) = d−1 ∑
k=0
b k φ(2x − k). If
∞∫
x=−∞
all k, then d−1 ∑
c i b i−2k = 0 for all k.
Proof:
∫ ∞
i=0
x=−∞
φ(x)ψ(x − k)dx =
φ(x)φ(x − k)dx = δ(k) and
∫ ∞
x=−∞ i=0
Interchanging the order of integration and summation
∑d−1
∑d−1
∫ ∞
c i b j
i=0
j=0
Substituting y = 2x − i yields
i=0
x=−∞
1 ∑d−1
∑d−1
∫ ∞
c i b j
2
j=0
y=−∞
∞∫
x=−∞
k=0
φ(x)ψ(x − k)dx = 0 for
∑d−1
∑d−1
c i φ(2x − i) b j φ(2x − 2k − j)dx = 0.
j=1
φ(2x − i)φ(2x − 2k − j)dx = 0
φ(y)φ(y − 2k − j + i)dy = 0
Thus,
Summing over j gives
∑d−1
∑d−1
c i b j δ(2k + j − i) = 0
i=0 j=0
∑d−1
c i b i−2k = 0
i=0
Lemma 11.5 gave a condition on the coefficients in the equations for φ(x) and ψ(x) if
integer shifts of the mother wavelet are to be orthogonal to the scale function. In addition,
for integer shifts of the mother wavelet to be orthogonal to the scale function requires
that b k = (−1) k c d−1−k .
Lemma 11.6 Let the scale function φ(x) equal d−1 ∑
ψ(x) equal d−1 ∑
k=0
k=0
b k φ(2x − k). If the scale functions are orthogonal
∫ ∞
−∞
φ(x)φ(x − k)dx = δ(k)
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c k φ(2x−k) and let the wavelet function