AI 中文总结
研究基于无条件收敛性质刻画具有广义平移不变结构的两两正交框架，给出包括Gabor等系统正交性条件，推导GTI Parseval框架刻画并显式构造系统对，改进技术，还举例说明结果及在采样理论中的应用。
AI 中文摘要
本文基于无条件收敛性质（UCP），对具有广义平移不变（GTI）结构的两两正交框架进行了刻画。这些GTI系统是通过在局部紧阿贝尔（LCA）群\(G\)的可数个闭的、余紧子群族上平移函数生成的，每个系统相关的子群族可能不同。作为该刻画的应用，建立了包括LCA群上的Gabor、小波和剪切波系统等各种结构化系统正交性的充要条件。还推导了GTI Parseval（紧）框架的刻画，并给出了使用滤波器的GTI系统对的显式构造。构造的系统满足\(\infty\)-UCP且卡尔德隆和等于1，构成Parseval框架且两两正交。该构造通过放宽子群族的平稳假设改进了文献\cite{RGS}中的技术。最后用\(B\)-样条作为生成函数举例说明结果，并讨论了两两正交框架在采样理论中的应用。
英文摘要
In this paper, we provide a characterization of pairwise orthogonal frames with generalized translation-invariant (GTI) structures, based on the unconditional convergence property (UCP). These GTI systems are generated by translating functions over a countable family of closed, co-compact subgroups of a locally compact abelian (LCA) group $G$, where the families of subgroups associated with each system may differ. As an application of this characterization, we establish necessary and sufficient criteria for the orthogonality of various structured systems, including Gabor, wavelet, and shearlet systems on LCA groups. Furthermore, we derive a characterization of GTI Parseval (tight) frames and present explicit constructions of pairs of GTI systems using filters. Each constructed system satisfies the $\infty$-UCP and admits a Calderón sum equal to one. As a consequence of these results, the constructed systems form Parseval frames and are pairwise orthogonal. The proposed construction improves upon the technique in \cite{RGS} by relaxing the stationary assumption on the families of subgroups. Finally, we illustrate the results with examples using $B$-splines as generating functions and discuss applications of pairwise orthogonal frames in sampling theory.
Comments41 pages
