Construction of Fractional Spline Wavelet Bases
We extend Schoenberg's B-splines to all fractional degrees Ī± > -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 Ī±-Hƶlder continuous for Ī± > 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 (Ī±+1), while they reproduce the polynomials of degree [Ī±]. We show how they yield continuous-order generalizations of the orthogonal Battle-LemariĆ© wavelets and of the semi-orthogonal B-spline wavelets. As Ī± 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.
1999
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431
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