In this work, we investigate and characterize linear functionals <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2175_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="189" /> </InlineMediaObject> <EquationSource Format="TEX">$L: \mathcal {V}\subsetneq \mathbb {R}[x_1,\dots ,x_n]\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>:</mo> <mi mathvariant="script">V</mi> <mo>⊊</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> with finite-dimensional <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2175_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation> and absolutely continuous representing measures <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2175_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, i.e., <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2175_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{d}\mu (x) = g(x)\,\textrm{d}x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>d</mtext> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="0.166667em" /> <mtext>d</mtext> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> for some density g. We focus on the regularity of the density g.

Absolutely continuous representing measures of moment functionals: the general truncated case

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2025年9月5日

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Philipp J. di Dio

摘要

In this work, we investigate and characterize linear functionals $L: \mathcal {V}\subsetneq \mathbb {R}[x_1,\dots ,x_n]\rightarrow \mathbb {R}$ $L : V ⊊ R [x_{1}, \dots, x_{n}] \to R$ with finite-dimensional $\mathcal {V}$ $V$ and absolutely continuous representing measures $\mu $ $μ$ , i.e., $\textrm{d}\mu (x) = g(x)\,\textrm{d}x$ $d μ (x) = g (x) d x$ for some density g. We focus on the regularity of the density g.