Foundations of Data Science

11.5 Conditions on the Dilation Equation
We would like a basis for a vector space of functions where each basis vector has
finite support and the basis vectors are orthogonal. This is achieved by a wavelet system
consisting of a shifted version of a scale function that satisfies a dilation equation along
with a set of wavelets of various scales and shifts. For the scale function to have a nonzero
integral, Lemma 11.1 requires that the coefficients of the dilation equation sum to two.
Although the scale function φ(x) for the Haar system has the property that φ(x) and
φ(x − k), k > 0, are orthogonal, this is not true for the scale function for the dilation
equation φ(x) = 1φ(2x) + φ(2x − 1) + 1 φ(2x − 2). The conditions that integer shifts of the
2 2
scale function be orthogonal and that the scale function has finite support puts additional
conditions on the coefficients of the dilation equation. These conditions are developed in
the next two lemmas.
Lemma 11.2 Let
∑d−1
φ(x) = c k φ(2x − k).
k=0
If φ(x) and φ(x − k) are orthogonal for k ≠ 0 and φ(x) has been normalized so that
∫ ∞
−∞ φ(x)φ(x − k)dx = δ(k), then ∑ d−1
i=0 c ic i−2k = 2δ(k).
Proof: Assume φ(x) has been normalized so that ∫ ∞
φ(x)φ(x − k)dx = δ(k). Then
−∞
Since
∫ ∞
x=−∞
∫ ∞
φ(x)φ(x − k)dx =
∫ ∞
x=−∞ i=0
∑d−1
∑d−1
∫ ∞
= c i c j
i=0
j=0
x=−∞
φ(2x − i)φ(2x − 2k − j)dx = 1 2
∑d−1
∑d−1
c i φ(2x − i) c j φ(2x − 2k − j)dx
= 1 2
x=−∞
∫ ∞
x=−∞
∫ ∞
x=−∞
j=0
= 1 δ(2k + j − i),
2
φ(2x − i)φ(2x − 2k − j)dx
φ(y − i)φ(y − 2k − j)dy
φ(y)φ(y + i − 2k − j)dy
∫ ∞
x=−∞
−∞
i=0
j=0
∑d−1
∑d−1
1
φ(x)φ(x − k)dx = c i c j
2 δ(2k + j − i) = 1 ∑d−1
c i c i−2k . Since φ(x) was nor-
2
malized so that
∫ ∞
∑d−1
φ(x)φ(x − k)dx = δ(k), it follows that c i c i−2k = 2δ(k).
i=0
i=0
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