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Abstract and Applied Analysis
Volume 2017 (2017), Article ID 3795120, 11 pages
https://doi.org/10.1155/2017/3795120
Research Article

A Variation on Uncertainty Principle and Logarithmic Uncertainty Principle for Continuous Quaternion Wavelet Transforms

1Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia
2Division of Mathematical Sciences, Osaka Kyoiku University, Osaka 582-8582, Japan

Correspondence should be addressed to Ryuichi Ashino; pj.ca.ukioyk-akaso.cc@onihsa

Received 23 October 2016; Accepted 5 January 2017; Published 30 January 2017

Academic Editor: Lucas Jodar

Copyright © 2017 Mawardi Bahri and Ryuichi Ashino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The continuous quaternion wavelet transform (CQWT) is a generalization of the classical continuous wavelet transform within the context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to establish logarithmic uncertainty principles related to generalized transform.