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Linear Canonical Wavelet Transform in Quaternion Domains

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Abstract

The well-known quaternion algebra is a four-dimensional natural extension of the field of complex numbers and plays a significant role in various aspects of signal processing, particularly for representing signals wherein several instincts are to be controlled simultaneously. For efficient analysis of such quaternionic signals, we introduce the notion of linear canonical wavelet transform in quaternion domain by invoking the elegant convolution structure associated with the quaternion linear canonical transform. The preliminary analysis encompasses the study of fundamental properties of the proposed linear canonical wavelet transform in quaternion domain including the Rayleigh’s theorem, inversion formula and a characterization of the range. Subsequently, we formulate three uncertainty principles; viz, Heisenberg-type, logarithmic and local uncertainty inequalities associated with the linear canonical wavelet transform in quaternion domain.

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Acknowledgements

The first author is supported by SERB (DST), Government of India, 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, Aajaz A. Teali & Azhar Y. Tantary

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  1. Firdous A. Shah
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  2. Aajaz A. Teali
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  3. Azhar Y. Tantary
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Correspondence to Firdous A. Shah.

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Shah, F.A., Teali, A.A. & Tantary, A.Y. Linear Canonical Wavelet Transform in Quaternion Domains. Adv. Appl. Clifford Algebras 31, 42 (2021). https://doi.org/10.1007/s00006-021-01142-7

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