Harmonic Analysis and Applications

June 4-8, 2018


"Orthogonality of generalized translation invariant frame pairs for function spaces on LCA groups"

Gumber, Anupam

In this article, we provide necessary and sufficient conditions for the orthogonality of two Bessel families when such families have the form of generalized translation invariant (GTI) systems over a second countable locally compact abelian (LCA) group $G$. The work is motivated by the utility of a recent notion given by Jakobsen and Lemvig on GTI systems in $L^2(G)$, and the concept of the orthogonality (or strongly disjointness) of a pair of frames studied by Balan, Han, and Larson. Consequently, we deduce similar results for several function systems including the case of TI systems, and GTI systems on compact abelian groups. We apply our results to the Bessel families having wave-packet structure (combination of wavelet as well as Gabor structure), and hence a characterization for pairwise orthogonal wave-packet frame systems over LCA groups is obtained. In addition, we relate the well-established theory from literature with our results by observing several deductions in the context of wavelet and Gabor systems over LCA groups with $G=\mathbb{R}^d,\mathbb{Z}^d$, etc. This is a joint work with Niraj Kumar Shukla, India. Gumber_PostDoc India_ Strobl18 Poster.pdf

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