<p>As an extension of the integer Cauchy power to the real powers of a linear functional, the concept of the index of positivity of a linear functional <i>u</i> is defined as the supremum of all nonnegative real numbers <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1923_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for which <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1923_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$u^\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mi>λ</mi> </msup> </math></EquationSource> </InlineEquation> remains positive. Some properties of the index map are studied. We prove that the index of positivity of the Hermite linear functional <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1923_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1923_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">$\text {Ind}_p(\mathcal {H}) = +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Ind</mtext> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Finally, some illustrative examples of linear functionals whose index of positivity is also infinite are shown.</p>

Index of Positivity of Linear Functionals: The Hermite Case.

期刊论文
发表日期：

2025年7月7日

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Ridha Sfaxi,
Francisco Marcellán

摘要

As an extension of the integer Cauchy power to the real powers of a linear functional, the concept of the index of positivity of a linear functional u is defined as the supremum of all nonnegative real numbers $\lambda \ge 0$ $λ \geq 0$ for which $u^\lambda $ $u^{λ}$ remains positive. Some properties of the index map are studied. We prove that the index of positivity of the Hermite linear functional $\mathcal {H}$ $H$ is $\text {Ind}_p(\mathcal {H}) = +\infty $ ${Ind}_{p} (H) = + \infty$ . Finally, some illustrative examples of linear functionals whose index of positivity is also infinite are shown.