The Continuous Wavelet Transform in Sobolev Spaces Over Locally Compact Abelian Group and Its Approximation Properties

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DOI: https://doi.org/10.28924/2291-8639-21-2023-139
Published: Dec 21, 2023
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C. P. Pandey, M. M. Dixit, Mopi Ado, Sunil Kumar Singh, The Continuous Wavelet Transform in Sobolev Spaces Over Locally Compact Abelian Group and Its Approximation Properties, Int. J. Anal. Appl., 21 (2023), 139.

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C. P. Pandey, M. M. Dixit, Mopi Ado, Sunil Kumar Singh

Abstract

The Sobolev space over locally compact abelian group Hs(G) is defined and we extend the continuous wavelet transform to Sobolev space Hs(G) for arbitrary real s. This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces. Further, the asymptotic behaviour of the transforms of the L2 function for small scaling parameters is examined. In special cases, the wavelet transform converges to a generalized derivative of its argument. We also discuss the consequences for the discrete wavelet transform arising from this property. Numerical examples illustrate the main result.

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References

  1. M.M. Dixit, C.P. Pandey, D. Das, Generalized Continuous Wavelet Transform on Locally Compact Abelian Group, Adv. Inequal. Appl. 2019 (2019), 10. https://doi.org/10.28919/aia/4067.
  2. A.G. Farashahi, Cyclic Wave Packet Transform on Finite Abelian Groups of Prime Order, Int. J. Wavelets Multiresolut Inf. Process. 12 (2014), 1450041. https://doi.org/10.1142/s0219691314500416.
  3. P. Gorka, T. Kostrzewa, E.G. Reyes, Sobolev Spaces on Locally Compact Abelian Groups: Compact Embeddings and Local Spaces, J. Funct. Spaces. 2014 (2014), 404738. https://doi.org/10.1155/2014/404738.
  4. A. Grossmann, M. Holschneider, R. Kronland-Martinet, J. Morlet, Detection of Abrupt Changes in Sound Signals With the Help of Wavelet Transforms, in: Inverse problems: an interdisciplinary study, Academic Press, London, pp. 289–306, (1987).
  5. A. Grossmann, J. Morlet, T. Paul, Transforms Associated to Square Integrable Group Representations. I. General Results, J. Math. Phys. 26 (1985), 2473–2479. https://doi.org/10.1063/1.526761.
  6. M. Holschneider, Wavelet Analysis Over Abelian Groups, Appl. Comput. Harmon. Anal. 2 (1995), 52–60. https://doi.org/10.1006/acha.1995.1004.
  7. C.P. Pandey, K. Moungkang, S.K. Singh, M.M. Dixit, M. Ado, Inversion Formula for the Wavelet Transform on Abelian Group, Int. J. Anal. Appl. 20 (2022), 64. https://doi.org/10.28924/2291-8639-20-2022-64.
  8. R.S. Pathak, A. Pathak, on Convolution for Wavelet Transform, Int. J. Wavelets Multiresolut Inf. Process. 06 (2008), 739–747. https://doi.org/10.1142/s0219691308002628.
  9. A. Ghaani Farashahi, Wave Packet Transform over Finite Fields, Electron. J. Linear Algebra. 30 (2015), 507–529. https://doi.org/10.13001/1081-3810.2903.
  10. A.G. Farashahi, Galois Wavelet Transforms Over Finite Fields, Rocky Mt. J. Math. 49 (2019), 79–99. https://doi.org/10.1216/rmj-2019-49-1-79.
  11. W. Rudin, Functional Analysis, McGraw Hill, New York, 1979.
  12. J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 2012.