Hardy’s uncertainty principle for Gabor transform on compact extensions of $$\mathbb {R}^n$$

Kais Smaoui
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Abstract

We prove in this paper a generalization of Hardy’s theorem for Gabor transform in the setup of the semidirect product \(\mathbb {R}^n\rtimes K\), where K is a compact subgroup of automorphisms of \(\mathbb {R}^n\). We also solve the sharpness problem and thus obtain a complete analogue of Hardy’s theorem for Gabor transform. The representation theory and Plancherel formula are fundamental tools in the proof of our results.

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在 $$\mathbb {R}^n$ 的紧凑扩展上进行 Gabor 变换的哈代不确定性原理
在本文中,我们证明了哈代定理在半间接积 \(\mathbb {R}^n\rtimes K\) 的设置中对 Gabor 变换的概括,其中 K 是 \(\mathbb {R}^n\) 的一个紧凑的自动子群。我们还解决了尖锐性问题,从而得到了哈代定理关于 Gabor 变换的完整类比。表示理论和 Plancherel 公式是证明我们结果的基本工具。
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