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Biorthogonal Wavelets on the Spectrum
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  • Owais Ahmad,
  • Neyaz Sheikh,
  • Kottakkaran Nisar,
  • Firdous Shah
Owais Ahmad
National Institute of Technology Srinagar
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Neyaz Sheikh
National Institute of Technology Srinagar
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Kottakkaran Nisar
Prince Sattam bin Abdulaziz University, Prince Sattam bin Abdulaziz University
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Firdous Shah
University of Kashmir
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Abstract

In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum $\Lambda=\left\{0, r/N\right\}+2\mathbb Z$, where $N\ge 1$ is an integer and $r$ is an odd integer with $1\le r\le 2N-1$ such that $r$ and $N$ are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families. Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

Peer review status:Published

21 Sep 2020Submitted to Mathematical Methods in the Applied Sciences
22 Sep 2020Submission Checks Completed
22 Sep 2020Assigned to Editor
30 Oct 2020Reviewer(s) Assigned
06 Nov 2020Review(s) Completed, Editorial Evaluation Pending
06 Nov 2020Editorial Decision: Revise Minor
07 Nov 20201st Revision Received
07 Nov 2020Submission Checks Completed
07 Nov 2020Assigned to Editor
07 Nov 2020Editorial Decision: Accept
20 Nov 2020Published in Mathematical Methods in the Applied Sciences. 10.1002/mma.7046