Construction of pairwise orthogonal Parseval frames generated by filters on LCA groups

Abstract

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 $L^2\left(G\right)$ 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 $L^2\left(R\right)$ and $L^2\left(G\right)$, using B-splines as a generating function. In the end, the results are particularly discussed for wavelet systems.