The generalized translation invariant (GTI) systems unify the discrete frame theory of generalized shift-invariant systems with its continuous version, such as wavelets, shearlets, Gabor transforms, and others. This article provides sufficient conditions to construct pairwise orthogonal Parseval GTI frames in L2(G) satisfying the local integrability condition (LIC) and having the Calderón sum one, where G is a second countable locally compact abelian group. The pairwise orthogonality plays a crucial role in multiple access communications, hiding data, synthesizing superframes and frames, etc. Further, we provide a result for constructing N numbers of GTI Parseval frames, which are pairwise orthogonal. Consequently, we obtain an explicit construction of pairwise orthogonal Parseval frames in L2(R) and L2(G), using B-splines as a generating function. In the end, the results are particularly discussed for wavelet systems.

中文翻译：

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构建由 LCA 组上的过滤器生成的成对正交 Parseval 帧


广义平移不变量 （GTI） 系统将广义移位不变量系统的离散框架理论与其连续版本（例如小波、剪切小波、Gabor 变换等）统一起来。本文提供了足够的条件，在 L2（G） 中构建成对的正交 Parseval GTI 帧，满足局部可积性条件 （LIC） 并具有 Calderón 和 1，其中 G 是第二个可数的局部紧凑阿贝尔群。成对正交性在多址通信、隐藏数据、合成超帧和帧等中起着至关重要的作用。此外，我们提供了构建 N 个 GTI Parseval 帧的结果，这些帧是成对正交的。因此，我们使用 B 样条作为生成函数，在 L2（R） 和 L2（G） 中获得了成对正交 Parseval 帧的显式构造。最后，特别讨论了小波系统的结果。