The two-scale relation for the uniform B-spline blending function can be used to represent this function as a linear combination of scaled and translated versions of itself. This remarkable property is extremely useful in defining wavelets on B-splines.
In these notes, we develop the coefficients of the linear combination. The fact that the blending function can be defined using convolution allows us to analyze this relationship in terms of its Fourier transform.
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The Two-Scale Relation for Uniform B-Splines
Given the general B-Spline blending function of order , the two-scale relation is written as
We calculate these coefficients by taking the Fourier transform of both sides of the two-scale equation.
Calculating the Fourier Transform of the Blending Function
First let be the Fourier Transform of , that is
Taking the Fourier Transform of Both Sides of the Two-Scale Equation
If we take the Fourier Transform of both sides of the equation
Comparing both sides of the above equation, we can see that