Complex B-Splines

We propose a complex generalization of Schoenberg's cardinal splines. To this end, we go back to the Fourier domain definition of the B-splines and extend it to complex-valued degrees. We show that the resulting complex B-splines are piecewise modulated polynomials, and that they retain most of the important properties of the classical ones: smoothness, recurrence, and two-scale relations, Riesz basis generator, explicit formulae for derivatives, including fractional orders, etc. We also show that they generate multiresolution analyses of $L ^{ 2 }$ (R) and that they can yield wavelet bases. We characterize the decay of these functions which are no-longer compactly supported when the degree is not an integer. Finally, we prove that the complex B-splines converge to modulated Gaussians as their degree increases, and that they are asymptotically optimally localized in the time-frequency plane in the sense of Heisenberg's uncertainty principle.