Designing local orthogonal bases on finite groups. I: Abelian case
Starting from the well-known equivalence property for the orthogonality of integer translates of functions \(f\in L_2\) \[ \langle f,f(\cdot-n)\rangle= \delta_{0n}\Leftrightarrow \sum_{K\in\mathbb{Z}} |\widehat f(\cdot+ 2\pi k)|^2= 1\quad\text{a.e.}\tag{1} \] the authors investigate a similar relation for general finite Abelian groups of unitary transforms. More precisely, they prove a similar orthogonality condition as in (1) for the sets \(\{Ub:U\in \Gamma\}\), \((b\in V)\), where \(V\) is a vector space with inner product and \(\Gamma\) is a finite Abelian group of unitary transforms on \(V\). The authors use their examinations to design an orthogonalization procedure for \(\{Ub: U\in\Gamma\}\). Further, they deal with several generating vectors \(b_1,\dots, b_L\). The non-Abelian case is addressed in Part II of this paper [ibid. 6, 207-231 (2000; see the following review)].
scientific article; zbMATH DE number 1445623