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<p>We establish sufficient conditions under which the HRT conjecture holds. Assuming a linear dependence among time-frequency shifts of a function in either the Schwartz class or the Wiener amalgam space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10181_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\( W_0(\mathbb {R}) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we analyze the associated structure via the Zak transform. We show that the zero set of the Zak transform is invariant under a specific translation, and that its algebraic properties depend on the rational dimension of the time-frequency parameters. In particular, if the rational dimension is three, the function must vanish identically, confirming the conjecture in this case.</p>

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Letter to the Editor: On a Special Configuration for the HRT Conjecture

  • 期刊论文

  • 发表日期：

    2025年6月20日

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  • Kasso A. Okoudjou,
  • Vignon Oussa

摘要

We establish sufficient conditions under which the HRT conjecture holds. Assuming a linear dependence among time-frequency shifts of a function in either the Schwartz class or the Wiener amalgam space \( W_0(\mathbb {R}) \) W 0 ( R ) , we analyze the associated structure via the Zak transform. We show that the zero set of the Zak transform is invariant under a specific translation, and that its algebraic properties depend on the rational dimension of the time-frequency parameters. In particular, if the rational dimension is three, the function must vanish identically, confirming the conjecture in this case.