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Spline Wavelets with Fractional Order of Approximation

M. Unser, T. Blu

Wavelet Applications Workshop, Monte Verità TI, Swiss Confederation, September 28-October 2, 1998.



We extend Schoenberg's family of polynomial splines with uniform knots to all fractional degrees α>-1/2. These splines, which involve linear combinations of the one sided power functions x+α=max{0,x}α, are α-Hölder continuous for α≥0. We construct the corresponding B-splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains. We show that these functions satisfy most of the properties of the traditional B-splines, including the convolution property, and a generalized fractional differentiation rule that involves finite differences only. We characterize the decay of the fractional B-splines which are not compactly supported for non-integral α's.


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