Construction of Fractional Spline Wavelet Bases

We extend Schoenberg's B-splines to all fractional degrees α > -1/2. These splines are constructed using linear combinations of the integer shifts of the power functions $x _{ + } ^{ \alpha }$ (one-sided) or $|x| _{ * } ^{ \alpha }$ (symmetric); in each case, they are α-Hölder continuous for α > 0. They satisfy most of the properties of the traditional B-splines; in particular, the Riesz basis condition and the two-scale relation, which makes them suitable for the construction of new families of wavelet bases. What is especially interesting from a wavelet perspective is that the fractional B-splines have a fractional order of approximation (α+1), while they reproduce the polynomials of degree [α]. We show how they yield continuous-order generalizations of the orthogonal Battle-Lemarié wavelets and of the semi-orthogonal B-spline wavelets. As α increases, these latter wavelets tend to be optimally localized in time and frequency in the sense specified by the uncertainty principle. The corresponding analysis wavelets also behave like fractional differentiators; they may therefore be used to whiten fractional Brownian motion processes.