Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1656_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\{z_n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> be a sequence in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1656_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {D}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation>. We give a sufficient and necessity condition that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1656_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\{z_n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is an interpolating sequence for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1656_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {Q}}_K\bigcap H^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">Q</mi> <mi>K</mi> </msub> <mo>⋂</mo> <msup> <mi>H</mi> <mi>∞</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. Moreover, this paper constructs an analytic f that <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1656_Article_IEq8.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="282" /> </InlineMediaObject> <EquationSource Format="TEX">$f(z_n)=\sum _j\lambda _jf_{z_j}(z_n)=\lambda _n, n=1,2,\ldots $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∑</mo> <mi>j</mi> </msub> <msub> <mi>λ</mi> <mi>j</mi> </msub> <msub> <mi>f</mi> <msub> <mi>z</mi> <mi>j</mi> </msub> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1656_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\lambda _n\}\in l^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mo>∈</mo> <msup> <mi>l</mi> <mi>∞</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, where f and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1656_Article_IEq10.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$f_{z_j}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <msub> <mi>z</mi> <mi>j</mi> </msub> </msub> </math></EquationSource> </InlineEquation> belong to <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1656_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {Q}}_K\bigcap H^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">Q</mi> <mi>K</mi> </msub> <mo>⋂</mo> <msup> <mi>H</mi> <mi>∞</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. We improve a result due to Nicolau and Xiao.

A Note of the Interpolating Sequences for ${\mathcal {Q}}_K\bigcap H^\infty $

期刊论文
发表日期：

2025年1月15日

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Jizhen Zhou,
Qingqing Wang

摘要

Let $\{z_n\}$ ${z_{n}}$ be a sequence in ${\mathbb {D}}$ $D$ . We give a sufficient and necessity condition that $\{z_n\}$ ${z_{n}}$ is an interpolating sequence for ${\mathcal {Q}}_K\bigcap H^\infty $ $Q_{K} ⋂ H^{\infty}$ . Moreover, this paper constructs an analytic f that $f(z_n)=\sum _j\lambda _jf_{z_j}(z_n)=\lambda _n, n=1,2,\ldots $ $f (z_{n}) = \sum_{j} λ_{j} f_{z_{j}} (z_{n}) = λ_{n}, n = 1, 2, \dots$ for any $\{\lambda _n\}\in l^\infty $ ${λ_{n}} \in l^{\infty}$ , where f and $f_{z_j}$ $f_{z_{j}}$ belong to ${\mathcal {Q}}_K\bigcap H^\infty $ $Q_{K} ⋂ H^{\infty}$ . We improve a result due to Nicolau and Xiao.

A Note of the Interpolating Sequences for \({\mathcal {Q}}_K\bigcap H^\infty \)

摘要