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<p>This paper investigates the orthogonality of Gabor frames generated by time-frequency shifts along model sets, a structured class of irregular translates arising in harmonic analysis. We establish necessary and sufficient conditions under which two such Gabor systems form a pair of orthogonal frames in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1831_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\( L^2(\mathbb R^n) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, characterized by the vanishing of the mixed dual Gramian operator. The results extend the classical theory of orthogonal Gabor frames beyond lattice structures and provide a constructive framework for analyzing time-frequency systems indexed by model sets. Additionally, we explore the dual frame property in the context of super Hilbert spaces formed by orthogonal direct sums, highlighting its implications for Gabor-type constructions.</p>

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Orthogonal Gabor Frames for Model Sets

  • 期刊论文

  • 发表日期：

    2025年10月12日

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  • Sudipta Sarkar

摘要

This paper investigates the orthogonality of Gabor frames generated by time-frequency shifts along model sets, a structured class of irregular translates arising in harmonic analysis. We establish necessary and sufficient conditions under which two such Gabor systems form a pair of orthogonal frames in \( L^2(\mathbb R^n) \) L 2 ( R n ) , characterized by the vanishing of the mixed dual Gramian operator. The results extend the classical theory of orthogonal Gabor frames beyond lattice structures and provide a constructive framework for analyzing time-frequency systems indexed by model sets. Additionally, we explore the dual frame property in the context of super Hilbert spaces formed by orthogonal direct sums, highlighting its implications for Gabor-type constructions.