Abstract
In this paper we consider the new concept of the quaternionic unitary representation of a locally compact group to the unitary group of a quaternionic Hilbert space, and study its properties. Also, we establish a continuous wavelet transform by means of a special case of such representations to extend the continuous wavelet transform related to semidirect product groups. The results may be useful in development of wavelet theory.
Similar content being viewed by others
References
Ali, S.T., Antoine, J.-P., Gazeau, J.-P.: Coherent States, Wavelets, and Their Generalization. Springer, New York (2000)
Alpay, D., Colombo, F., Gantner, J., Kimsey, David P.: Functions of the Infinitesimal Generator of a Strongly Continuous Quaternionic Group. arXiv:1502.02954 (2015)
Antoine, J.-P., Murenzi, R., Vandergheynst, P., Twareque Ali, S.: Two Dimensional Wavelets and Their Relatives. Cambridge University Press, Cambridge (2004)
Bahri, M., Ashino, R., Vaillancourt, R.: Continuous quaternion Fourier and wavelet transforms. Int. J. Wavelets Multiresolut. Inf. Process. 12, 1460003 (2014)
Bayro-Corrochano, E.: The theory and use of the quaternion wavelet transform. J. Math. Imaging Vis. 24, 1935 (2006)
Colombo, F., Gantner, J, Janssen, T.: Schatten Class and Berezin Transform of Quaternionic Linear Operators. arXiv:1511.07308v1 (2015)
Fashandi, M., Kamyabi Gol, R.A., Niknam, A., Pourabdollah, M.A.: Continuous wavelet transform on a special homogeneous space. J. Math. Phys. 44, 4260–4266 (2003)
Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Inc., Boca Raton (1995)
Ghiloni, R., Moretti, V., Perotti, A.: Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys. 25, 1350006 (2013)
Ghiloni, R., Moretti, V., Perotti, A.: Spectral properties of compact normal quaternionic operators. In: Bernstein, S., Kähler, U., Sabadini, I., Sommen, F. (eds.) Hypercomplex Analysis: New Perspectives and Applications, Trends in Mathematics, pp. 133–143. Birkhauser, Basel (2014)
Ghiloni, R., Moretti, V., Perotti, A.: Spectral Representations of Normal Operators in Quaternionic Hilbert Spaces via Intertwining Quaternionic PVMS. arXiv:1602.02661 (2016)
He, J., Yu, B.: Continuous wavelet transform on the space \(L^2(\mathbf{R}, \mathbb{H}, dx)\). Appl. Math. Lett. 17, 111–121 (2004)
Thirulogasanthar, K., Muraleetharan, B.: Coherent States on Quaternion Slices and a Measurable Field of Hilbert Spaces. arXiv:1603.02509v1 (2016)
Twareque Ali, S., Thirulogasanthar, K.: The quaternionic affine group and related continuous wavelet transforms on complex and quaternionic Hilbert spaces. J. Math. Phys. 55, 063501 (2014)
Twareque Ali, S.: On Some Quaternionic Coherent States and Wavelets. Geometric Methods in Physics. XXXIII Workshop 2014, Trends in Mathematics, 103–111 (2015)
Viswanath, K.: Normal operators on quaternionic Hilbert spaces. Trans. Am. Math. Soc. 162, 337–350 (1971)
Acknowledgements
The author would like to express the most sincere gratitude to the memory of Professor Syed Twareque Ali from Concordia University, for the valuable discussions on the initial version of the manuscript. Also, the author would like to thank the reviewers for their comments and suggestions on the paper. This research was supported by a grant from Ferdowsi University of Mashhad; No. 2/43973.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fashandi, M. Quaternionic continuous wavelet transform on a quaternionic Hilbert space. RACSAM 112, 1049–1057 (2018). https://doi.org/10.1007/s13398-017-0409-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-017-0409-4