Quaternionic Fundamental Cardinal Splines: Interpolation and Sampling

Abstract

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 $1/\sum\limits_{k\in\Z} \widehat{B}_{q}(\xi+2 \pi k)$---if well-defined---in the frequency domain yields a cardinal fundamental spline of quaternionic order that does satisfy the interpolation property. We handle the ambiguity of the quaternion-valued exponential function appearing in the denominator of the interpolation filter and relate the filter to interesting properties of a quaternionic Hurwitz zeta function and the existence of complex quaternionic inverses. Finally, we show that the cardinal fundamental splines of quaternionic order fit into the setting of Kramer's Lemma and allow for a family of sampling, respectively, interpolation series.