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<p>The aim of this paper is to investigate the right-sided quaternionic free metaplectic transformation (QFMT) and its associated uncertainty principles (UPs) for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1125_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^{2d}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>-dimensional quaternionic-valued signals. First, we establish the fundamental mathematical properties of the QFMT, including partial derivatives, the inversion formula, Parseval’s theorem, and the Hausdorff–Young inequality. Next, we establish various UPs within this framework, such as the Rènyi and Shannon entropy UPs and Donoho–Stark’s UP in terms of concentration. Finally, we derive <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13324_2025_1125_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^a\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>a</mi> </msup> </math></EquationSource> </InlineEquation>-bandlimited variant of the Donoho–Stark UP in the QFMT domain.</p>

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The right-sided quaternionic free metaplectic transformation and associated uncertainty principles

  • 期刊论文

  • 发表日期：

    2025年9月3日

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  • Khaled Hleili,
  • Youssef El Haoui

摘要

The aim of this paper is to investigate the right-sided quaternionic free metaplectic transformation (QFMT) and its associated uncertainty principles (UPs) for \(\mathbb {R}^{2d}\) R 2 d -dimensional quaternionic-valued signals. First, we establish the fundamental mathematical properties of the QFMT, including partial derivatives, the inversion formula, Parseval’s theorem, and the Hausdorff–Young inequality. Next, we establish various UPs within this framework, such as the Rènyi and Shannon entropy UPs and Donoho–Stark’s UP in terms of concentration. Finally, we derive \(L^a\) L a -bandlimited variant of the Donoho–Stark UP in the QFMT domain.