Foundations of Data Science

11.7 Sufficient Conditions for the Wavelets to be Orthogonal
Section 11.6 gave necessary conditions on the b k and c k in the definitions of the scale
function and wavelets for certain orthogonality properties. In this section we show that
these conditions are also sufficient for certain orthogonality conditions. One would like a
wavelet system to satisfy certain conditions.
1. Wavelets, ψ j (2 j x − k), at all scales and shifts to be orthogonal to the scale function
φ(x).
2. All wavelets to be orthogonal. That is
∫ ∞
−∞
ψ j (2 j x − k)ψ l (2 l x − m)dx = δ(j − l)δ(k − m)
3. φ(x) and ψ jk , j ≤ l and all k, to span V l , the space spanned by φ(2 l x − k) for all k.
These items are proved in the following lemmas. The first lemma gives sufficient conditions
on the wavelet coefficients b k in the definition
ψ(x) = ∑ k
b k ψ(2x − k)
for the mother wavelet so that the wavelets will be orthogonal to the scale function. That
is, if the wavelet coefficients equal the scale coefficients in reverse order with alternating
negative signs, then the wavelets will be orthogonal to the scale function.
Lemma 11.7 If b k = (−1) k c d−1−k , then ∫ ∞
−∞ φ(x)ψ(2j x − l)dx = 0 for all j and l.
Proof: Assume that b k = (−1) k c d−1−k . We first show that φ(x) and ψ(x − k) are orthogonal
for all values of k. Then we modify the proof to show that φ(x) and ψ(2 j x − k) are
orthogonal for all j and k.
Assume b k = (−1) k c d−1−k . Then
∫ ∞
−∞
φ(x)ψ(x − k) =
∫ ∞
∑d−1
∑d−1
c i φ(2x − i) b j φ(2x − 2k − j)dx
−∞ i=0
j=0
∑d−1
∑d−1
∫ ∞
= c i (−1) j c d−1−j φ(2x − i)φ(2x − 2k − j)dx
i=0
j=0
−∞
∑d−1
∑d−1
= (−1) j c i c d−1−j δ(i − 2k − j)
i=0
j=0
∑d−1
= (−1) j c 2k+j c d−1−j
j=0
= c 2k c d−1 − c 2k+1 c d−2 + · · · + c d−2 c 2k−1 − c d−1 c 2k
= 0
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