Abstract
In this paper, we prove the Riemann-Lebesgue lemma for the Clifford-Fourier transform. We give an alternative proof of Heisenberg’s inequality. Furthermore, we provide a version of Young inequalities in the framework of Clifford analysis.
Similar content being viewed by others
References
Brackx, F., Hitzer, E., Sangwine, S.J.: History of quaternion and Clifford-Fourier transforms. In: Hitzer, E., Sangwine, S.J. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets, Trends in Mathematics (TIM), vol. 27, pp xi–xxvii. Birkhäuser, Basel (2013)
Batard, T., Berthier, M., Saint-Jean, C.: Clifford Fourier transform for color image processing. In: Bayro-Corrochano, E., Scheuermann, G. (eds.) Geometric Algebra Computing: In Engineering and Computer Science, pp 135–161. Springer, London (2010)
Brackx, F., De Schepper, N., Sommen, F.: The Clifford-Fourier transform. J. Fourier Anal. Appl. 11, 669–681 (2005)
Bujack, R., De Bie, H., De Schepper, N., Scheuermann, G.: Convolution products for hypercomplex Fourier transforms. J. Math. Imaging Vis. 48, 606–624 (2014)
Bujack, R., Scheuermann, G., Hitzer, E.: A general geometric fourier transform convolution theorem. Adv. Appl. Clifford Algebr. 23, 15–38 (2013)
De Bie, H., De Schepper, N., Sommen, F.: The class of Clifford-Fourier transforms. J. Fourier Anal. Appl. 17, 1198–1231 (2011)
De Bie, H., Oste, R., Van der Jeugt, J.: Generalized Fourier transforms arising from the enveloping algebras of s l(2) and osp(1—2). Int. Math. Res. Not. 15, 4649–4705 (2016)
De Bie, H., Xu, Y.: On the Clifford-Fourier transform. Int. Math. Res. Not. 22, 5123–5163 (2011)
Ebling, J., Scheuermann, G.: Clifford Fourier transform on vector fields. IEEE Trans. Vis. Comput. Graph 11, 469–479 (2005)
Ell, T.A., Sangwine, S.J.: Hypercomplex Fourier transforms of color images. IEEE Trans. Image Process 16, 22–35 (2007)
Ghobber, S., Jaming, P.H.: Uncertainty principles for integral operators. Studia Mathematica, INSTYTUT MATEMATYCZNY POLSKA AKADEMIA NAUK 220, 197–220 (2014)
Hitzer, E., Bahri, M.: Clifford Fourier transform on multivector fields and uncertainty principles for dimensions n = 2 (mod4) and n = 3 (mod4). Adv. Appl. Clifford Algebr. 18, 715–736 (2008)
Shimeno, N.: A note on the uncertainty principle for the Dunkl transform. J. Math. Sci. Univ. Tokyo 8, 33–42 (2001)
Sommen, F.: Special functions in clifford analysis and axial symmetry. J. Math. Anal. Appl. 130, 110–133 (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
El Kamel, J., Jday, R. Inequalities in the Setting of Clifford Analysis. Math Phys Anal Geom 21, 36 (2018). https://doi.org/10.1007/s11040-018-9295-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-018-9295-z