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Finite Dual \(\mathbf{g}\)-Framelet Systems Associated with an Induced Group Action

  • Anupam Gumber
  • Niraj K. Shukla
Article

Abstract

In this article, we first induce an action of a topological group \({\mathbb {G}}\) on \(\ell ^2({\mathbb {Z}}_N^d)\) from a given action of \({\mathbb {G}}\) on the space \({\mathbb {C}}\) of complex numbers. Then, for each \( \mathrm {g} \in {\mathbb {G}}\), we introduce a framelet system (\(\mathbf{g}\)-framelet system or \(\mathbf{g}\)-FS) associated with an induced action of \({\mathbb {G}}\) on \(\ell ^2({\mathbb {Z}}_N^d)\), and a super \(\mathbf{g}\)-FS for the super-space in the same set-up. By applying the group-theoretic approach based on the complete digit set, we characterize the generators of two \(\mathbf{g}\)-framelet systems (super \(\mathbf{g}\)-framelet systems) such that they form a \(\mathbf{g}\)-dual pair (super \(\mathbf{g}\)-dual pair). As a consequence, characterizations for the Parseval \(\mathbf{g}\)-FS and the Parseval super \(\mathbf{g}\)-FS are obtained. Further, some properties of the frame operator corresponding to the \(\mathbf{g}\)-FS are observed, which results in concluding that its canonical dual preserves the same structure.

Keywords

Wavelet system Framelet Dual framelet Frame operator Dual Gramian Group action 

Mathematics Subject Classification

42C40 42C15 

Notes

Acknowledgements

The authors would like to thank all anonymous reviewers concerned with this article. They also want to thank K. K. Azad and Ashisha Kumar for fruitful research discussions, and also Divya Singh and Rajeshwari Dubey for giving access to their unpublished work on induced group action and wavelet bases. The first named author gratefully acknowledges the Ministry of Human Resource Development, India for the research fellowship and Indian Institute of Technology Indore, India for the support provided during the period of this work.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Discipline of MathematicsIndian Institute of Technology IndoreSimrol, IndoreIndia

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