A useful inequality for quaternion linear canonical transform

Mawardi Bahri

doi:10.62918/hjma.v1i1.7

Authors

Mawardi Bahri Universitas Hasanuddin

DOI:

https://doi.org/10.62918/hjma.v1i1.7

Keywords:

quaternion Fourier transform, quaternion canonical linear transform, inequality

Abstract

In this work, we first introduce the quaternion Fourier transform. We explore its relation to the quaternion linear Fourier transform and utilize this fact to extend an inequality for the quaternion Fourier transform in the framework of the quaternion linear canonical transform.

References

A. Abouelaz, A. Achak, R. Daher, N. Safouane, Donoho-Stark’s uncertainty principlefor the quaternion Fourier transform. Bol. Soc. Mat. Mex. 26 (2020) 587–597.

Achak, A. Abouelaz, R. Daher, N. Safouane, Uncertainty principles for the quaternion linear canonical transform, Adv. App. Clifford Algebr. 29 (2019) 99.

M. Bahri, M. Saleh, Relation between quaternion Fourier transform and quaternion Wigner-Ville distribution associated with linear canonical transform, J. Appl. Math. 2017 (2017) 3247364.

M. Bahri, Quaternion algebra-valued wavelet transform, Appl. Math. Sci. 5 (2011) 3531-3540.

M. Bahri, A. Lawi, N. Aris, M. Saleh, M. Nur, Relationships between convolution and correlation for Fourier transform and quaternion Fourier transform, Int. J. Math. Anal. 7 (2013) 2101-2109.

M. Bahri, S. A. Karim, A variation on inequality for quaternion Fourier transform, modified convolution and correlation theorems for general quaternion linear canonical transform, Symmetry 14 (2022) 1303.

T. B ̈ulow, Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images,Ph.D. thesis, University of Kiel, Germany (1999).

L. P. Chen, K. I. Kou, M. S. Liu, Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform, J. Math. Anal. Appl. 423 (2015) 681–700.

W. B. Gao, B. Z. Li, Quaternion windowed linear canonical transform of two-dimensional signals, Adv. Appl. Clifford Algebr. 30 (2020) 16.

E. Hitzer, Quaternion Fourier transform on quaternion fields and generalizations, Adv. App. Clifford Algebr. 20 (2007) 497–517.

P. Lian, Uncertainty principle for the quaternion Fourier transform, J. Math. Anal. Appl. 467 (2018) 1258–1269.

P. Lian, Sharp Hausdorff-Young inequalities for the quaternion Fourier transforms, Proc. Am. Math. Soc. 148 (2020) 697–703.

Y. N. Zhang, B. Z. Li, Novel uncertainty principles for two-sided quaternion linear canonical transform, Adv. App. Clifford Algebr. 28 (2018) 15.

X. Zhu, S. Zheng, Uncertainty principles for the two-sided quaternion linear canonical transform, Circuits, Syst. Signal Process. 39 (2020) 4436–4458.

X. Zhu, S. Zheng, On uncertainty principle for the two-sided quaternion linear canonical transform, J. Pseudo-Differ. Oper. Appl. 12 (2021).