Unser, MichaelMichaelUnserTHIERRY BLU2024-03-082024-03-081999-12-010277786Xhttps://scholars.lib.ntu.edu.tw/handle/123456789/640693We 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+α (one-sided) or |x|*α (symmetric); in each case, they are α-Holder 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-Lemarie 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.conference paper2-s2.0-0033343879https://api.elsevier.com/content/abstract/scopus_id/0033343879