Fractional Splines and Wavelets
M. Unser, T. Blu
SIAM Review, vol. 42, no. 1, pp. 43–67, March 2000.
We extend Schoenberg's family of polynomial splines with uniform knots to all fractional degrees α > -1. These splines, which involve linear combinations of the one-sided power functions x+α = max(0, x)α, belong to L1 and 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 B-splines which are not compactly supported for non-integral α's. Their most astonishing feature (in reference to the Strang-Fix theory) is that they have a fractional order of approximation α + 1 while they reproduce the polynomials of degree [α]. For α > 1/2, they satisfy all the requirements for a multiresolution analysis of L2 (Riesz bounds, two scale relation) and may therefore be used to build new families of wavelet bases with a continuously-varying order parameter. Our construction also yields symmetrized fractional B-splines which provide the connection with Duchon's general theory of radial (m,s)-splines (including thin-plate splines). In particular, we show that the symmetric version of our splines can be obtained as solution of a variational problem involving the norm of a fractional derivative. (Front cover).
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