izvuzmathИзвестия высших учебных заведений. МатематикаIzvestiya Vysshikh Uchebnykh Zavedenii. Matematika0021-34462076-4626Казанский (Приволжский) федеральный университет10.26907/0021-3446-2025-2-61-73izvuzmath-163Research ArticleСтатьиОктонионное преобразование мориты в октонионных функциональных пространствахThe octonionic Moritoh transform on octonionic function spacesКумарА.KumarA.

Кумар Авния.

Лакхнау, 226025; Мотихари, Бихар, 845401

Awniya Kumar.

Lucknow, 226025; Motihari, Bihar, 845401

awniyakumar554@gmail.comСингхС. К.SinghS. K.

Сингх Сунил Кумар.

Лакхнау, 226025

Sunil Kumar Singh.

Lucknow, 226025

sks_math@yahoo.comСингхШ. К.SinghSh. K.

Сингх Шео Кумар.

Мотихари, Бихар, 845401

Sheo Kumar Singh.

Motihari, Bihar, 845401

sheokumarsingh@mgcub.ac.inУниверситет Бабасахеба Бхимрао Амбедкара; Центральный университет Махатмы ГандиИндияBabasaheb Bhimrao Ambedkar University; Mahatma Gandhi Central UniversityIndiaУниверситет Бабасахеба Бхимрао АмбедкараИндияBabasaheb Bhimrao Ambedkar UniversityIndiaЦентральный университет Махатмы ГандиИндияMahatma Gandhi Central UniversityIndia202526022025026173Copyright © Кумар А., Сингх С.К., Сингх Ш.К., 20252025Кумар А., Сингх С.К., Сингх Ш.К.Kumar A., Singh S.K., Singh S.K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://izvuzmath.elpub.ru/jour/article/view/163

Исследуются свойства и приложения октонионного преобразования Фурье с акцентом на вейвлет-анализе. Вейвлет-преобразование Мориты расширяется на октонионные пространства Бесова, взвешенные октонионные пространства Бесова, октонионные пространства BMO и взвешенные октонионные пространства BMO. Полученные оценки для октонионного вейвлет-преобразования Мориты в этих пространствах способствуют более глубокому пониманию его поведения. Наши результаты открывают перспективы для будущих исследований в обработке сигналов, анализе изображений и взаимодействии октонионного вейвлет-анализа с другими математическими теориями.

This article explores the challenges and applications of the octonion Fourier transform, with a focus on wavelet analysis. We extend the Moritoh wavelet transform to the octonionic Besov spaces, the weighted octonionic Besov spaces, the octonionic BMO spaces, and the octonionic weighted BMO spaces. The derived bounds for the octonionic Moritoh wavelet transform in these spaces contribute to a deeper understanding of its behaviour. Our findings pave the way for future research in signal processing, image analysis, and the intersection of octonion wavelet analysis with other mathematical theories.

октонионное преобразование Фурьевейвлет-преобразованиепространство Бесовапространство BMOoctonion Fourier transformwavelet transformBesov spaceBMO spaceReferencesLian P. The octonionic Fourier transform: Uncertainty relations and convolution, Signal Process. 164 (2), 295–300 (2019), DOI: 10.1016/j.sigpro.2019.06.015.Lian P. The octonionic Fourier transform: Uncertainty relations and convolution, Signal Process. 164 (2), 295–300 (2019), DOI: 10.1016/j.sigpro.2019.06.015.Błaszczyk Ł., Snopek K.M. Octonion Fourier transform of real-valued functions of three variables-selected properties and examples, Signal Process. 136, 29–37 (2017), DOI: 10.1016/j.sigpro.2016.11.021.Błaszczyk Ł., Snopek K.M. Octonion Fourier transform of real-valued functions of three variables-selected properties and examples, Signal Process. 136, 29–37 (2017), DOI: 10.1016/j.sigpro.2016.11.021.Heredia C.J., Garcia E.A. Espinosa C.V. One dimensional octonion Fourier transform, J. Math. Control. Sci. Appl. 7 (1), 91–106 (2021).Heredia C.J., Garcia E.A. Espinosa C.V. One dimensional octonion Fourier transform, J. Math. Control. Sci. Appl. 7 (1), 91–106 (2021).Błaszczyk Ł. Octonion Spectrum of 3D Octonion-Valued Signals Properties and Possible Applications, 2018 26th European Signal Processing Conf. (EUSIPCO), IEEE, 2018, DOI: 10.23919/EUSIPCO.2018.8553228.Błaszczyk Ł. Octonion Spectrum of 3D Octonion-Valued Signals Properties and Possible Applications, 2018 26th European Signal Processing Conf. (EUSIPCO), IEEE, 2018, DOI: 10.23919/EUSIPCO.2018.8553228.Katunin A. Three-dimensional octonion wavelet transform, J. Appl. Math. Comput. Mech. 13 (1), 33–38 (2014).Katunin A. Three-dimensional octonion wavelet transform, J. Appl. Math. Comput. Mech. 13 (1), 33–38 (2014).Moritoh S. Wavelet transforms in Euclidean spaces their relation with wave front sets and Besov, Triebel-Lizorkin spaces, Tohoku Math. J. 47 (4) (2), 555–565 (1995), DOI: 10.2748/tmj/1178225461.Moritoh S. Wavelet transforms in Euclidean spaces their relation with wave front sets and Besov, Triebel-Lizorkin spaces, Tohoku Math. J. 47 (4) (2), 555–565 (1995), DOI: 10.2748/tmj/1178225461.Baez J.C. The octonions, Bull. Amer. Math. Soc. 39 (2), 145–205 (2002), DOI: 10.1090/S0273-0979-01-00934-X.Baez J.C. The octonions, Bull. Amer. Math. Soc. 39 (2), 145–205 (2002), DOI: 10.1090/S0273-0979-01-00934-X.Axler S. Measure, Integration & Real Analysis (Springer Nature, 2020), DOI: 10.1007/978-3-030-33143-6.Axler S. Measure, Integration & Real Analysis (Springer Nature, 2020), DOI: 10.1007/978-3-030-33143-6.Kumar A., Singh S.K., Singh S.K. A Note on Moritoh Transforms, Creat. Math. and Inform. 33 (2), 185–201 (2024), DOI: 10.37193/CMI.2024.02.05.Kumar A., Singh S.K., Singh S.K. A Note on Moritoh Transforms, Creat. Math. and Inform. 33 (2), 185–201 (2024), DOI: 10.37193/CMI.2024.02.05.Hörmander L. The Analysis of Linear Partial Differential Operators II (Springer-Verlag, Berlin Heidelberg, New York, Tokyo, 1983).Hörmander L. The Analysis of Linear Partial Differential Operators II (Springer-Verlag, Berlin Heidelberg, New York, Tokyo, 1983).Singh S.K., Kalita B. The S-transform on Hardy spaces and its duals, Int. J. Anal. Appl. 7 (2), 171–178 (2015).Singh S.K., Kalita B. The S-transform on Hardy spaces and its duals, Int. J. Anal. Appl. 7 (2), 171–178 (2015).Chuong N.M., Duong D.V. Boundedness of the wavelet integral operator on weighted function spaces, Russ. J. Math. Phys. 20, 268–275 (2013), DOI: 10.1134/S1061920813030023.Chuong N.M., Duong D.V. Boundedness of the wavelet integral operator on weighted function spaces, Russ. J. Math. Phys. 20, 268–275 (2013), DOI: 10.1134/S1061920813030023.Yin Ming, Liu Wei, Shui Jun, Wu Jiangmin Quaternion Wavelet Analysis and Application in Image Denoising, Math. Probl. Engineering, 493976 (2012), DOI: 10.1155/2012/493976.Yin Ming, Liu Wei, Shui Jun, Wu Jiangmin Quaternion Wavelet Analysis and Application in Image Denoising, Math. Probl. Engineering, 493976 (2012), DOI: 10.1155/2012/493976.Bayro-Corrochano E. The Theory and Use of the Quaternion Wavelet Transform, J. Math. Imaging Vis. 24 (1), 19–35 (2006), DOI: 10.1007/s10851-005-3605-3.Bayro-Corrochano E. The Theory and Use of the Quaternion Wavelet Transform, J. Math. Imaging Vis. 24 (1), 19–35 (2006), DOI: 10.1007/s10851-005-3605-3.Chan W.L. Choi H., Baraniuk R. Quaternion wavelets for image analysis and processing, 2004 Intern. Conf. on Image Processing, 2004. ICIP’04 5, 3057–3060, IEEE, 2004, DOI: 10.1109/ICIP.2004.1421758.Chan W.L. Choi H., Baraniuk R. Quaternion wavelets for image analysis and processing, 2004 Intern. Conf. on Image Processing, 2004. ICIP’04 5, 3057–3060, IEEE, 2004, DOI: 10.1109/ICIP.2004.1421758.Chan W.L., Choi H., Baraniuk R.G. Coherent Multiscale Image Processing Using Dual-Tree Quaternion Wavelets, IEEE Transactions on Image Processing 17 (7), 1069–1082 (2008), DOI: 10.1109/TIP.2008.924282.Chan W.L., Choi H., Baraniuk R.G. Coherent Multiscale Image Processing Using Dual-Tree Quaternion Wavelets, IEEE Transactions on Image Processing 17 (7), 1069–1082 (2008), DOI: 10.1109/TIP.2008.924282.Lhamu D., Singh S.K. The quaternion Fourier and wavelet transforms on spaces of functions and distributions, Res. Math. Sci. 7 (11) (2020), DOI: 10.1007/s40687-020-00209-4.Lhamu D., Singh S.K. The quaternion Fourier and wavelet transforms on spaces of functions and distributions, Res. Math. Sci. 7 (11) (2020), DOI: 10.1007/s40687-020-00209-4.Lhamu D., Singh S.K., Pandey C.P. The continuous quaternion wavelet transform on function spaces, Boletim da Sociedade Paranaense de Matem. 42 (2024), DOI: 10.5269/bspm.63502.Lhamu D., Singh S.K., Pandey C.P. The continuous quaternion wavelet transform on function spaces, Boletim da Sociedade Paranaense de Matem. 42 (2024), DOI: 10.5269/bspm.63502.Singh S.K. Besov norms in terms of the S-transform, Afrika Matem. 27 (3), 603–615 (2015), DOI: 10.1007/s13370-015-0365-0.Singh S.K. Besov norms in terms of the S-transform, Afrika Matem. 27 (3), 603–615 (2015), DOI: 10.1007/s13370-015-0365-0.Lhamu D, Singh S.K. Besov norms of the continuous wavelet transform in variable Lebesgue space, J. Pseudo-Differ. Oper. Appl. 11 (3), 1537–1548 (2020), DOI: 10.1007/s11868-020-00361-z.Lhamu D, Singh S.K. Besov norms of the continuous wavelet transform in variable Lebesgue space, J. Pseudo-Differ. Oper. Appl. 11 (3), 1537–1548 (2020), DOI: 10.1007/s11868-020-00361-z.

The authors declare that there are no conflicts of interest present.