复空间和四元数Hilbert空间上的四元数仿射群及相关的连续小波变换
The Quaternionic Affine Group and Related Continuous Wavelet Transforms on Complex and Quaternionic Hilbert Spaces
摘要：
类比于实仿射群和复仿射群的酉不可约表示用于定义一维和二维连续小波变换，本文研究了四元数仿射群，并构造了它的酉不可约表示。这些表示都是在复希尔伯特空间和四元数希尔伯特空间上构造的。在实数和复数的情况下，四元数群的表示也是平方可积的。利用这些表示，我们在复希尔伯特空间和四元数希尔伯特空间上构造了四元数小波和连续小波变换。
英文摘要：
By analogy with the real and complex affine groups, whose unitary irreducible representations are used to define the one and two-dimensional continuous wavelet transforms, we study here the quaternionic affine group and construct its unitary irreducible representations. These representations are constructed both on a complex and a quaternionic Hilbert space. As in the real and complex cases, the representations for the quaternionic group also turn out to be square-integrable. Using these representations we constrct quaternionic wavelets and continuous wavelet transforms on both the complex and quaternionic Hilbert spaces.
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