On the orthogonality of frame pairs over locally compact abelian groups

Abstract

The theory of frames has a very close connection with unitary group representations, since various structured systems, for example, wavelet and Gabor systems can be realized as a sequence obtained by applying a family of unitary operators to a particular window function. Among various frame properties, the \orthogonality or strongly disjointness"of a pair of frames for Hilbert spaces is a very useful concept introduced and studied by Balan, Han, and Larson. A pair of frames satisfying the above mentioned property are termed as pairwise orthogonal (simply, orthogonal). Orthogonal frames play a key role in frame theory, for example, in construction of new frames from existing ones, constructions related with duality, in multiple access communications, hiding data, and synthesizing super-frames, etc.In this thesis we study the orthogonality of a pair of frames for function systems generated by regular representations of locally compact abelian (LCA) groups. To the best of our knowledge, we realize that the characterization results for pairwise orthogonal frames have not been studied earlier in the context of LCA groups, and are appearing rst time in the literature via the work presented in this thesis. Precisely, we examine pairwise orthogonality of frames generated by the action of a unitary representation of a countable family of closed and co-compact subgroups f􀀀jgj2J G on a separable Hilbert space L2(G), where J Z and G is a second countable LCA group. We consider frames of the form E( ) := n ( ) p;j : p;j 2 ; 2 􀀀j ; p 2 Pj ; j 2 J ofor a (not necessarily countable) family := f p;jgp2Pj ;j2J in L2(G). We pay special attention to this problem by assuming as the action of a countable family f􀀀jgj2J on L2(G) by (left-)translation. The representation of 􀀀 acting on L2(G) by (left-)translation is called the (left-)regular representation of 􀀀.We obtain characterizations of orthogonal frame pairs of the form E( ) by considering di erent situations on a countable family of closed and co-compact subgroups f􀀀jgj2J in G. Along with this, the resulting characterizations are used to construct new frames and super-frames by using various techniques including the unitary extension principle by Ron and Shen and its recent extension to LCA groups by Christensen and Goh.Additionally, we investigate the orthogonality of structured function systems (e.g., wavelet, Gabor, and wave-packet systems) over LCA-group setting. Further, we relate the above-developed theory to the classical case as well as nite dimensional Hilbert spaces by letting LCA group G of the form Rd;Zd; and Zd N; etc.Lastly, we provide a group-theoretic construction of a nite frame wavelet (simply, framelet) system associated with an induced group action. Along with this, we investigate the time-frequency localization properties of the framelet system. We also study the orthogonality and duality properties of a pair of framelet systems, and its applications in synthesizing super framelet systems over nite dimensional set-up.