Quaternionic B-splines

We introduce B-splines Bq of quaternionic order q, defined on the real line for the purposes of multi-channel signal analysis. The functions Bq are defined first by their Fourier transforms, then as the solutions of a distributional differential equation of quaternionic order. The equivalence of these definitions requires properties of quaternionic Gamma functions and binomial expansions, both of which we investigate. The relationship between Bq and a backwards difference operator is shown, leading to a recurrence formula. We show that the collection of integer shifts of Bq is a Riesz basis for its span, hence generating a multiresolution analysis. Finally, we demonstrate the pointwise and Lp convergence of the quaternionic B-splines to quaternionic Gaussian functions.