- Title
- Quaternionic B-splines
- Creator
- Hogan, Jeffrey A.; Massopust, Peter
- Relation
- Funding BodyARCGrant NumberDP160101537 http://purl.org/au-research/grants/arc/DP160101537
- Relation
- Journal of Approximation Theory Vol. 224, p. 43-65
- Publisher Link
- http://dx.doi.org/10.1016/j.jat.2017.09.003
- Publisher
- Academic Press
- Resource Type
- journal article
- Date
- 2017
- Description
- 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.
- Subject
- quaternions; B-splines; Clifford algebra; quaternionic binomial; quaternionic Gamma function; multiresolution analysis
- Identifier
- http://hdl.handle.net/1959.13/1384436
- Identifier
- uon:32073
- Identifier
- ISSN:0021-9045
- Language
- eng
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