# Quaternionic Fundamental Cardinal Splines: Interpolation and Sampling

@article{Hogan2019QuaternionicFC,
title={Quaternionic Fundamental Cardinal Splines: Interpolation and Sampling},
author={Jeffrey A. Hogan and Peter R. Massopust},
journal={Complex Analysis and Operator Theory},
year={2019}
}
• Published 18 April 2018
• Mathematics
• Complex Analysis and Operator Theory
B-splines $B_{q}$, $\Sc q > 1$, of quaternionic order $q$, for short quaternionic B-splines, are quaternion-valued piecewise M\"{u}ntz polynomials whose scalar parts interpolate the classical Schoenberg splines $B_{n}$, $n\in\N$, with respect to degree and smoothness. As the Schoenberg splines of order $\geq 3$, they in general do not satisfy the interpolation property $B_{q}(n-k) = \delta_{n,k}$, $n,k\in\Z$. However, the application of the interpolation filter $1/\sum\limits_{k\in\Z} \widehat… 2 Citations ## Figures from this paper On Some Generalizations of B-Splines In this article, we consider some generalizations of polynomial and exponential B-splines. Firstly, the extension from integral to complex orders is reviewed and presented. The second generalization Interpolation and Sampling with Exponential Splines of Real Order We establish the existence of fundamental cardinal exponential B-splines of positive real order$\sigma$subject to two conditions on$\sigma$and implement their construction. A sampling result for ## References SHOWING 1-10 OF 26 REFERENCES Fractional Splines and Wavelets • Mathematics, Computer Science SIAM Rev. • 2000 The symmetric version of the B-splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative, and may be used to build new families of wavelet bases with a continuously varying order parameter. Quaternionic B-splines • Mathematics, Computer Science J. Approx. Theory • 2017 The relationship between quaternionic B-splines and a backwards difference operator is shown, leading to a recurrence formula, and it is shown that the collection of integer shifts of$B_q\$ is a Riesz basis for its span, hence generating a multiresolution analysis.
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