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The Mexican Hat Wavelet Transform on Generalized Quotients and Its Applications

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Advances in Mathematical Modelling, Applied Analysis and Computation (ICMMAAC 2022)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 666))

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Abstract

The theory of wavelet analysis is used to characterize functions and distribution spaces intrinsically. It is a field that is constantly evolving and is a mathematical approach widely used for many applications. Recently, the theory of Mexican hat wavelet transform (MHWT) on distributions and its properties are derived by Pathak et al. [10]. Further, Singh et al. [18] constructed Representation theorems for the same transform with some applications.

In this chapter, we study the Mexican hat wavelet transform (MHWT) to the space of generalized quotients with its operational properties and applications. We extend MHWT as a continuous linear map between the spaces of generalized quotients. An inversion and a characterization theorem for the MHWT of generalized quotients are also discussed. Further, Mexican hat wavelet transformation is defined on the space of tempered generalized quotients by employing the structure of exchange property. We study the exchange property for the Mexican hat wavelet transform by applying the theory of the Mexican hat wavelet transform of distributions. Furthermore, different properties of Mexican hat wavelet transform are discussed on the space of tempered generalized quotients with applications.

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Acknowledgement

The first author (AS) is supported by National Board for Higher Mathematics(DAE), Government of India, through sanction No. 02011/7/2022 NBHM(R.P.)/R &D-II/10010 and the Second author is supported by DST under WOS-A, Government of India.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Banasthali Vidhyapith, Banasthali, India

    Abhishek Singh

  2. Department of Physics, Banaras Hindu University, Varanasi, India

    Shubha Singh

  3. Department of Mathematics and Statistics, Rajasthan University, Jaipur, India

    Devendra Kumar

Authors
  1. Abhishek Singh
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  2. Shubha Singh
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  3. Devendra Kumar
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Corresponding author

Correspondence to Abhishek Singh .

Editor information

Editors and Affiliations

  1. Department of Mathematics, JECRC University, Jaipur, Rajasthan, India

    Jagdev Singh

  2. Department of Mathematics, University of Memphis, Memphis, TN, USA

    George A. Anastassiou

  3. Department of Mathematics, Cankaya University, Institute of space sciences, Bucharest, Ankara, Türkiye

    Dumitru Baleanu

  4. Department of Mathematics, University of Rajasthan, Jaipur, India

    Devendra Kumar

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Singh, A., Singh, S., Kumar, D. (2023). The Mexican Hat Wavelet Transform on Generalized Quotients and Its Applications. In: Singh, J., Anastassiou, G.A., Baleanu, D., Kumar, D. (eds) Advances in Mathematical Modelling, Applied Analysis and Computation . ICMMAAC 2022. Lecture Notes in Networks and Systems, vol 666. Springer, Cham. https://doi.org/10.1007/978-3-031-29959-9_5

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  • DOI: https://doi.org/10.1007/978-3-031-29959-9_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-29958-2

  • Online ISBN: 978-3-031-29959-9

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