Splines of complex order
Complex B-splines, defined in Forster, Blu & Unser (Appl. Comp. Harmon. Anal., 20: 281–282, 2006) are a natural extension of the classical Curry-Schoenberg (polynomial) B-splines and the fractional splines introduced by Unser & Blu. (Fractional splines and wavelets. SIAM Review, 42(1): 43–67, 2000.) Although these complex B-splines have properties that are similar to those of the classical polynomial B-splines, they in addition provide a means of extracting more information from a signal or image by employing the imaginary part or phase.
In collaboration with B. Forster at the Technische Universität München, we have exhibited relationships between complex B-splines, fractional derivatives and integrals, and Dirichlet averages, that allow a more general complex B-spline to be defined. An extension to the multivariate setting and the generalization of known properties of classical multivariate polynomial B-splines was also achieved. Recently, we introduced a new class of splines of complex order by considering a class of fractional differential operators of complex order. These new splines include the (cardinal) complex B-splines are a special case. In addition, we derived a sampling theorem for a class of splines of complex order that connects to properties of the Hurwitz zeta function and constructed periodic splines of complex order.
Publications:
1. B. Forster and P. Massopust, Some Remarks about the Connection between Fractional Divided Differences, Fractional B-Splines, and the Hermite-Genocchi Formula, International Journal of Wavelets, Multiresolution and Information Processing, Vol. 6, No. 2, 279 – 290 (2008).
2. B. Forster and P. Massopust, Statistical Encounters with Complex B-Splines, Constr. Approx. 29, 325 – 344 (2009).
3. B. Forster and P. Massopust, Multivariate Complex B-Splines, Dirichlet Averages and Difference Operators, Proceedings of SampTA 2009.
4. P. Massopust, Double Dirichlet Averages and Complex B-Splines, Proceedings of SampTA 2009.
5. P. Massopust and B. Forster, Multivariate Complex B-splines and Dirichlet Averages, J. Approx. Theory 162, 252 – 269 (2010).
6. B. Forster and P. Massopust, Splines of Complex Order: Fourier, Filter, and Fractional Derivatives, to appear in Sampling Theory in Signal and Image Analysis.