Foundations of Data Science

Calculate φ(2x−1), φ(2x−2), and φ(2x−3) by shifting the contents
of φ(2x) the appropriate distance, discard the zeros that move off
the right end and add zeros at the left end.
Calculate the new approximation for φ(x) using the above values
for φ(2x − 1), φ(2x − 2), and φ(2x − 3) in the dilation equation for
φ(2x).
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Figure 11.3: Daubechies scale function and associated wavelet
The convergence of the iterative procedure for computing is fast if the eigenvectors of
a certain matrix are unity.
Another approach to solving the dilation equation
Consider the dilation equation φ(x) = 1f(2x) + f(2x − 1) + 1 f(2x − 2) and consider
2 2
continuous solutions with support in 0 ≤ x < 2.
φ(0) = 1φ(0) + φ(−1) + φ(−2) = 1 φ(0) + 0 + 0 φ(0) = 0
2 2
φ(2) = 1φ(4) + φ(3) + φ(2) = 1 φ(2) + 0 + 0 φ(2) = 0
2 2
φ(1) = 1 φ(2) + φ(1) + φ(0) = 0 + φ(1) + 0 φ(1) arbitrary
2
Set φ(1) = 1. Then
φ( 1 2 ) = 1 2 φ(1) + φ(0) + 1 2 φ(−1) = 1 2
φ( 3 2 ) = 1 2 φ(3) + φ(2) + 1 2 φ(1) = 1 2
Etc.
φ( 1 4 ) = 1 2 φ( 1 2 ) + φ(− 1 2 ) + 1 2 φ(− 3 2 ) = 1 4
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