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

• Published in: Applied and computational harmonic analysis Vol. 74; p. 101708
• Main Authors: Redhu, Navneet, Gumber, Anupam, Shukla, Niraj K.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-01-2025
• Subjects: Generalized translation invariant systems; Local integrablity condition; Locally compact abelian groups; Pairwise orthogonal frames; Parseval frames; Wavelets and B-splines; Pairwise orthogonal frames; Parseval frames; 42C40; Local integrablity condition; Generalized translation invariant systems; 42C15; 65T60; Locally compact abelian groups; Wavelets and B-splines; 43A70
• DOI: 10.1016/j.acha.2024.101708
• ISSN: 1063-5203

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.