Let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_728_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}L^q_s ({\textbf{R}}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <msubsup> <mi>L</mi> <mi>s</mi> <mi>q</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="bold">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the set of all tempered distributions <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_728_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$f \in \mathcal {S}^\prime ({\textbf{R}}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="bold">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> such that the norm <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_728_Article_IEq5.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="306" /> </InlineMediaObject> <EquationSource Format="TEX">$\Vert f \Vert _{\mathcal {F}L^q_s} := \left( \int _{{\textbf{R}}^2} \left( |\mathcal {F}[f](\xi )| (1+|\xi |)^s \right) ^q d\xi \right) ^{1/q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <mi mathvariant="script">F</mi> <msubsup> <mi>L</mi> <mi>s</mi> <mi>q</mi> </msubsup> </mrow> </msub> <mo>:</mo> <mo>=</mo> <msup> <mfenced close=")" open="("> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="bold">R</mi> </mrow> <mn>2</mn> </msup> </msub> <msup> <mfenced close=")" open="("> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="script">F</mi> <mrow> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mo stretchy="false">(</mo> <mn>1</mn> </mrow> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>ξ</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> </mfenced> <mi>q</mi> </msup> <mi>d</mi> <mi>ξ</mi> </mfenced> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>q</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> is finite, where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_728_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}[f]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the Fourier transform of f. We investigate the spectral synthesis for the unit circle <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_728_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$S^1 \subset {\textbf{R}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="bold">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_728_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}L^q_s ({\textbf{R}}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <msubsup> <mi>L</mi> <mi>s</mi> <mi>q</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="bold">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_728_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$1<q<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_728_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$s> 2 / q^\prime $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>2</mn> <mo stretchy="false">/</mo> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_728_Article_IEq11.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$q^\prime $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> denotes the conjugate exponent of q.

On the spectral synthesis for the unit circle in $\mathcal {F} L_s^q ({\textbf{R}}^2)$

期刊论文
发表日期：

2025年8月20日

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Masaharu Kobayashi,
Enji Sato

摘要

Let $\mathcal {F}L^q_s ({\textbf{R}}^2)$ $F L_{s}^{q} (R^{2})$ denote the set of all tempered distributions $f \in \mathcal {S}^\prime ({\textbf{R}}^2)$ $f \in S^{'} (R^{2})$ such that the norm $\Vert f \Vert _{\mathcal {F}L^q_s} := \left( \int _{{\textbf{R}}^2} \left( |\mathcal {F}[f](\xi )| (1+|\xi |)^s \right) ^q d\xi \right) ^{1/q}$ ${‖ f ‖}_{F L_{s}^{q}} : = {(\int_{R^{2}}, {(| F [f] (ξ) | (1 + {| ξ |)}^{s})}^{q}, d, ξ)}^{1 / q}$ is finite, where $\mathcal {F}[f]$ $F [f]$ denotes the Fourier transform of f. We investigate the spectral synthesis for the unit circle $S^1 \subset {\textbf{R}}^2$ $S^{1} \subset R^{2}$ in $\mathcal {F}L^q_s ({\textbf{R}}^2)$ $F L_{s}^{q} (R^{2})$ with $1<q<\infty $ $1 < q < \infty$ and $s> 2 / q^\prime $ $s > 2 / q^{'}$ , where $q^\prime $ $q^{'}$ denotes the conjugate exponent of q.

On the spectral synthesis for the unit circle in \(\mathcal {F} L_s^q ({\textbf{R}}^2)\)

摘要