<p>In this article, the convolution theorems and continuity results for the linear canonical curvelet transform (LCCT) are established. Using the generalized translation operator, the convolution theorem associated with the LCCT is formulated, and some new results are presented. Spectral and spatial convolution theorems are also derived. Furthermore, the LCCT is extended to function spaces such as generalized Sobolev, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_511_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-Sobolev, and Besov spaces. The approximation property of the LCCT is examined in the generalized Sobolev space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_511_Article_IEq2.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$B^{w,M}_{p,k}({\mathbb {R}}^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>k</mi> </mrow> <mrow> <mi>w</mi> <mo>,</mo> <mi>M</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. It is shown that the LCCT is a continuous linear operator on Sobolev spaces <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_511_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">${H}^{M}_{s}({\mathbb {R}}^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mi>s</mi> <mi>M</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40574_2025_511_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">${H}^{M}_{s,p}({\mathbb {R}}^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi>H</mi> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>p</mi> </mrow> <mi>M</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Additionally, continuity and boundedness results for the LCCT within the weighted Besov space are presented.</p>

The linear canonical curvelet transform on function spaces

期刊论文
发表日期：

2025年10月7日

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Soumya Singh,
Sunil Kumar Singh,
Karm Veer Singh

摘要

In this article, the convolution theorems and continuity results for the linear canonical curvelet transform (LCCT) are established. Using the generalized translation operator, the convolution theorem associated with the LCCT is formulated, and some new results are presented. Spectral and spatial convolution theorems are also derived. Furthermore, the LCCT is extended to function spaces such as generalized Sobolev, $L^p$ $L^{p}$ -Sobolev, and Besov spaces. The approximation property of the LCCT is examined in the generalized Sobolev space $B^{w,M}_{p,k}({\mathbb {R}}^{2})$ $B_{p, k}^{w, M} (R^{2})$ . It is shown that the LCCT is a continuous linear operator on Sobolev spaces ${H}^{M}_{s}({\mathbb {R}}^{2})$ $H_{s}^{M} (R^{2})$ and ${H}^{M}_{s,p}({\mathbb {R}}^{2})$ $H_{s, p}^{M} (R^{2})$ . Additionally, continuity and boundedness results for the LCCT within the weighted Besov space are presented.