<p>The Szegö-Dirichlet kernel of the right half-plane <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1735_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {H}}_{1/2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">H</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is given by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1735_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">${\varkappa }(s, u) = \zeta (s+{\overline{u}}),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϰ</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mover> <mi>u</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1735_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$s, u \in {\mathbb {H}}_{1/2},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>,</mo> <mi>u</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">H</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1735_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation> denotes the Riemann zeta function. We show that none of the positive integer powers of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1735_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\varkappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϰ</mi> </math></EquationSource> </InlineEquation> has the 2-point scalar Pick property. Nevertheless, a network realization formula for the right half-plane <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1735_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {H}}_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">H</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is obtained.</p>

Nevanlinna-Pick Interpolation in the Right Half-Plane

期刊论文
发表日期：

2025年6月20日

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Sameer Chavan,
Chaman Kumar Sahu

摘要

The Szegö-Dirichlet kernel of the right half-plane ${\mathbb {H}}_{1/2}$ $H_{1 / 2}$ is given by ${\varkappa }(s, u) = \zeta (s+{\overline{u}}),$ $ϰ (s, u) = ζ (s + \bar{u}),$ $s, u \in {\mathbb {H}}_{1/2},$ $s, u \in H_{1 / 2},$ where $\zeta $ $ζ$ denotes the Riemann zeta function. We show that none of the positive integer powers of $\varkappa $ $ϰ$ has the 2-point scalar Pick property. Nevertheless, a network realization formula for the right half-plane ${\mathbb {H}}_0$ $H_{0}$ is obtained.