Quaternionic B-splines

We introduce B-splines B-q of quaternionic order q, defined on the real line for the purposes of multi-channel signal analysis. The functions B-q 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 B-q and a backwards difference operator is shown, leading to a recurrence formula. We show that the collection of integer shifts of B-q is a Riesz basis for its span, hence generating a multiresolution analysis. Finally, we demonstrate the pointwise and L-P convergence of the quaternionic B-splines to quaternionic Gaussian functions. (C) 2017 Elsevier Inc. All rights reserved.