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<p>The periodic Wigner transform is introduced. We show that most of the properties of the Euclidean Wigner transform are satisfied in this new setting. Using the periodic Wigner transform, we define the periodic Weyl transform. <i>L</i><sup>2</sup>-boundedness of periodic Weyl transforms are investigated. We give a necessary and sufficient condition on the symbol to ensure that the corresponding periodic Weyl transform is a Hilbert-Schmidt operator. We show that the product of two periodic Weyl transforms and the adjoint of a periodic Weyl transform are again periodic Weyl transforms. The connection between pseudo-differential operators on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10587_2025_6924_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{S}^{1}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and periodic Weyl transforms is given.</p>

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Periodic analogs of Wigner transforms and Weyl transforms

  • 期刊论文

  • 发表日期：

    2025年3月3日

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  • Shahla Molahajloo,
  • Man Wah Wong

摘要

The periodic Wigner transform is introduced. We show that most of the properties of the Euclidean Wigner transform are satisfied in this new setting. Using the periodic Wigner transform, we define the periodic Weyl transform. L2-boundedness of periodic Weyl transforms are investigated. We give a necessary and sufficient condition on the symbol to ensure that the corresponding periodic Weyl transform is a Hilbert-Schmidt operator. We show that the product of two periodic Weyl transforms and the adjoint of a periodic Weyl transform are again periodic Weyl transforms. The connection between pseudo-differential operators on \(\mathbb{S}^{1}\) S 1 and periodic Weyl transforms is given.