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A family of convolution-based generalized Stockwell transforms

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Abstract

The main purpose of this paper is to introduce a family of convolution-based generalized Stockwell transforms in the context of time-fractional-frequency analysis. The spirit of this article is completely different from two existing studies (see D. P. Xu and K. Guo [Appl. Geophys. 9 (2012) 73–79] and S. K. Singh [J. Pseudo-Differ. Oper. Appl. 4 (2013) 251–265]) in the sense that our approach completely relies on the convolution structure associated with the fractional Fourier transform. We first study all of the fundamental properties of the generalized Stockwell transform, including a relationship between the fractional Wigner distribution and the proposed transform. In the sequel, we introduce both the semi-discrete and discrete counterparts of the proposed transform. We culminate our investigation by establishing some Heisenberg-type inequalities for the generalized Stockwell transform in the fractional Fourier domain.

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Acknowledgements

The authors would like to thank the esteemed editor and the referee for their valuable comments and suggestions. The second-named 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 and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

    H. M. Srivastava

  2. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan, 40402, Republic of China

    H. M. Srivastava

  3. Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007, Baku, Azerbaijan

    H. M. Srivastava

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

    Firdous A. Shah & Azhar Y. Tantary

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  1. H. M. Srivastava
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  2. Firdous A. Shah
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  3. Azhar Y. Tantary
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Correspondence to Firdous A. Shah.

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Srivastava, H.M., Shah, F.A. & Tantary, A.Y. A family of convolution-based generalized Stockwell transforms. J. Pseudo-Differ. Oper. Appl. 11, 1505–1536 (2020). https://doi.org/10.1007/s11868-020-00363-x

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