Novel Gabor-Type Transform and Weighted Uncertainty Principles

Saifallah Ghobber () and Hatem Mejjaoli
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Saifallah Ghobber: Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
Hatem Mejjaoli: Department of Mathematics, College of Sciences, Taibah University, P.O. Box 30002, Al Madinah AL Munawarah 42353, Saudi Arabia

Mathematics, 2025, vol. 13, issue 7, 1-37

Abstract: The linear canonical Fourier transform is one of the most celebrated time-frequency tools for analyzing non-transient signals. In this paper, we will introduce and study the deformed Gabor transform associated with the linear canonical Dunkl transform (LCDT). Then, we will formulate several weighted uncertainty principles for the resulting integral transform, called the linear canonical Dunkl-Gabor transform (LCDGT). More precisely, we will prove some variations in Heisenberg’s uncertainty inequality. Then, we will show an analog of Pitt’s inequality for the LCDGT and formulate a Beckner-type uncertainty inequality via two approaches. Finally, we will derive a Benedicks-type uncertainty principle for the LCDGT, which shows the impossibility of a non-trivial function and its LCDGT to both be supported on sets of finite measure. As a side result, we will prove local uncertainty principles for the LCDGT.

Keywords: Dunkl transform; linear canonical transform; generalized translation; generalized convolution; uncertainty principles (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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