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<p>We prove in this paper an analogue of Hardy’s theorem for the Gabor transform in the setup of the semi-direct product <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K\ltimes {\mathbb {H}_n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>⋉</mo> <msub> <mi mathvariant="double-struck">H</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <i>K</i> is a compact subgroup of the group of automorphisms of the Heisenberg group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathbb {H}_n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">H</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. The representation theory and Plancherel formula are the fundamental tools in the proofs.</p>

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Hardy’s theorem for Gabor transform on compact extensions of the Heisenberg group

  • 期刊论文

  • 发表日期：

    2025年10月9日

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  • Mounir Elloumi

摘要

We prove in this paper an analogue of Hardy’s theorem for the Gabor transform in the setup of the semi-direct product \(K\ltimes {\mathbb {H}_n}\) K ⋉ H n , where K is a compact subgroup of the group of automorphisms of the Heisenberg group \({\mathbb {H}_n}\) H n . The representation theory and Plancherel formula are the fundamental tools in the proofs.