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Nonuniform multiresolution analysis associated with linear canonical transform

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Abstract

The linear canonical transform (LCT) has attained respectable status within a short span and is being broadly employed across several disciplines of science and engineering including signal processing, optical and radar systems, electrical and communication systems, quantum physics etc, mainly due to the extra degrees of freedom and simple geometrical manifestation. In this article, we introduce a novel multiresolution analysis (LCT-NUMRA) on the spectrum \(\Omega = \big \{0,{r}/{N}\big \}+2\mathbb {Z},\) where \(N \geqq 1\) is an integer and r is an odd integer with \( 1 \leqq r \leqq 2N-1,\) such that r and N are relatively prime, by intertwining the ideas of nonuniform MRA and linear canonical transforms. We first develop nonuniform multiresolution analysis associated with the LCT and then drive an algorithm to construct an LCT-NUMRA starting from a linear canonical low-pass filter \(\Lambda _{0}^{M}(\omega )\) with suitable conditions. Nevertheless, to extend the scope of the present study, we construct the associated wavelet packets for such an MRA and investigate their properties by means of the linear canonical transforms.

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Acknowledgements

The authors would like to thank the esteemed editor and the referee for their valuable comments and suggestions. The first author was financially supported by the Science and Engineering Research Board, Department of Science and Technology, Government of India under Grant No. EMR/2016/007951.

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Authors and Affiliations

  1. Department of Mathematics, University of Kashmir, South Campus, Anantnag, Jammu and Kashmir, 192101, India

    Firdous A. Shah & Waseem Z. Lone

  2. Department of Mathematics, College of Sciences, Taibah University, P.O. Box 30002, Al Madinah, Al Munawarah, Saudi Arabia

    Hatem Mejjaoli

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  1. Firdous A. Shah
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  2. Waseem Z. Lone
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  3. Hatem Mejjaoli
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Correspondence to Waseem Z. Lone.

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Shah, F.A., Lone, W.Z. & Mejjaoli, H. Nonuniform multiresolution analysis associated with linear canonical transform. J. Pseudo-Differ. Oper. Appl. 12, 21 (2021). https://doi.org/10.1007/s11868-021-00398-8

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  • DOI: https://doi.org/10.1007/s11868-021-00398-8

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