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Applicable Analysis

On fractional uncertainty: a dyadic approach

Hugo Aimar, P. G. Bolcatto, Ivana Gómez

Published Jun 21, 2019• 1 citation
Publisher

Abstract

The use of Haar wavelet analysis, instead of classical trigonometric analysis, provides inequalities which can be regarded as Heisenberg-type inequalities associated to fractional order quantum mechanics. The aim of this paper is to introduce and prove some of these lower order uncertainty estimates.
DOI
10.1080/00036811.2019.1629425

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It is well known that the fractional Fourier transform (FrFT) is a non-trivial generalization of the classical Fourier transform. This transform has been extensively utilized in signal analysis and image processing. The uncertainty principle is one of the fundamental results related to the FrFT. In the present paper, we first propose the different method in proving a variant of Heisenberg uncertainty principles associated with the FrFT. This method combines the Heisenberg-type uncertainty principle pertaining to the FrFT and the additive property of the FrFT. We then provide an alternative proof of the logarithmic inequality for the FRFT and find that this result differs slightly that in the literature. Finally, we explicitly expand the variant of Heisenberg uncertainty principle to a more general case.
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