• Korean Journal of Mathematics
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• v.23 no.1
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• pp.129-151
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• 2015

In this paper, we prove an $L^p$ version of Donoho-Stark's uncertainty principle for the inverse of the hypergeometric Fourier transform on $\mathbb{R}^d$. Next, using the ultracontractive properties of the semigroups generated by the Heckman-Opdam Laplacian operator, we obtain an $L^p$ Heisenberg-Pauli-Weyl uncertainty principle for the inverse of the hypergeometric Fourier transform on $\mathbb{R}^d$.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.521-535
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• 2022

In this paper, we introduce the k-Hankel-Wigner transform on R in some problems of time-frequency analysis. As a first point, we present some harmonic analysis results such as Plancherel's, Parseval's and an inversion formulas for this transform. Next, we prove a Heisenberg's uncertainty principle and a Calderón's reproducing formula for this transform. We conclude this paper by studying an extremal function for this transform.

• Cross-Cultural Studies
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• v.34
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• pp.93-114
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• 2014

Gaston Bachelard had a revolutionary progress in the field of human understanding by proposing his theory of image and imagination. His theory of the new image was so powerful, almost all areas of human science, particularly that of literary criticism, were strongly influenced and this influence continues until today. Today almost everyone accepts his theory of the image without much objection, but not rarely asked where began his transfer from the philosophy of science to the images. We propose a hypothesis that the beginning of the new concept of Bachelard's image was inspired by studies of contemporary science, especially quantum mechanics. The Heisenberg's uncertainty principle was the core of quantum mechanics, and opens new perspectives on the material world. We could summarize the message of the uncertainty principle : the material world is made up of various layers, and the material can not be measured by the location and movement at the same time. So we must have a new point of view of another dimension to know this material world. Bachelard had accepted this view of Heisenberg and developed his own theory of epistemological rupture. What is revolutionary in the theory of Bachelard's image is the fact that he looked at the images with the new perspective. The human psyche is another world compared to the rational world that dominates our daily lives. Bachelard insists that the image can not be explained by the concept. The fantasy world is a totally different world to that of rationality. That is why it can not be explained by the language of rationality as the concept. The imaginary world exists independently of the real world, but it is superimposed on the real world. These two worlds are influencing each other, and it is between these two world where our daily lives continues. The declaration of Bachelard 'image is a specific reality' is never a metaphor or rhetorical expression. This is an ontological expression that must truthfully. The imaginary world is a world built on the image and it works according to its own law. It is not a representation or copy of the real world. But the world of imagination are not alone. It exists in the same time and space with the world of science. It is superimposed with the world of science. Both two world influence each other. Bachelard has made a revolutionary change by studying the images. He gave them their own place. It has changed the views on the images that were treated as mere representations of reality. Thanks to him, the image can have its own value, that of a factor that creates reality. Bachelard shows how we can go deep into the source of being and the universe if we look at the pictures with the eyes of other dimensions.