Fractional Splines and Wavelets

Abstract

We extend Schoenberg's family of polynomial splines with uniform knots to all fractional degrees $\alpha-1$. These splines, which involve linear combinations of the one-sided power functions $x_{+}^{\alpha}=\max(0,x)^{\alpha}$, are $\alpha$-Hölder continuous for $\alpha0$. 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 that are not compactly supported for nonintegral $\alpha$'s. Their most astonishing feature (in reference to the Strang--Fix theory) is that they have a fractional order of approximation $\alpha+1$ while they reproduce the polynomials of degree $\lceil\alpha\rceil$. For $\alpha-\frac{1}{2}$, they satisfy all the requirements for a multiresolution analysis of $\LL^{2}$ (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 the solution of a variational problem involving the norm of a fractional derivative.