系统理论与正交多小波
System theory and orthogonal multi-wavelets
摘要：
本文给出了紧支撑单变量标量正交小波和向量值或矩阵值正交多小波的完整统一的刻画。这种刻画是基于系统理论和基本线性代数的经典结果。特别是，我们表明，相应的小波和多小波掩模的保守线性系的传递函数$$ F(z)=A+B z (I-Dz)^{-1} \, C, \quad z \in \mathbb{D}=\{z \in \mathbb{C} \ : \ |z| < 1\}, $$确定。复矩阵$A,\ B, \ C, \ D$定义块循环酉矩阵。我们的研究结果表明，有没有优雅的Daubechies的小波结构或任何其他向量值或矩阵值多小波的建设之间的本质区别。由$A,\ B, \ C, \ D$定义的酉矩阵的结构允许我们以系统的方式参数化所有类别的可能的小波和多小波掩模连同相应的可细化函数的掩模。
英文摘要：
In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function $$ F(z)=A+B z (I-Dz)^{-1} \, C, \quad z \in \mathbb{D}=\{z \in \mathbb{C} \ : \ |z| < 1\}, $$ of a conservative linear system. The complex matrices $A,\ B, \ C, \ D$ define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multi-wavelets. The structure of the unitary matrix defined by $A,\ B, \ C, \ D$ allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.
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