四元B样条
Quaternionic B-Splines
摘要：
为了多通道信号和图像分析的目的，我们在四元数阶$B_q$（四元数代数中的$q$）的线上引入B样条。函数$B_q$首先由它们的傅里叶变换定义，然后作为四元数阶分布微分方程的解。这些定义的等价性需要四元数Γ函数和二项式展开式的性质，这两个我们调查。给出了$B_q$和向后差分算子之间的关系，并导出了递推公式.我们发现，收集的整数移位的$B_q$是一个Riesz的基础上，其跨度，从而产生一个多分辨率分析。最后，我们证明了四元数B-样条到四元数高斯函数的逐点收敛性和$L^p$收敛性。
英文摘要：
We introduce B-splines on the line of quaternionic order $B_q$ ($q$ in the algebra of quaternions) for the purposes of multi-channel signal and image analysis. The functions $B_q$ are defined first by their Fourier transforms, then as the solutions of 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 quarternionic Gaussian functions.
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