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• W3044062023 abstract "Abstract In measure theory several results are known how measure spaces are transformed into each other. But since moment functionals are represented by a measure we investigate in this study the effects and implications of these measure transformations to moment funcationals, especially with dimensionality reduction. We gain characterizations of moment functionals. Among other things we show that for a compact and path connected set $$Ksubset mathbb {R}^n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> there exists a measurable function $$g:Krightarrow [0,1]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>:</mml:mo> <mml:mi>K</mml:mi> <mml:mo>→</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> such that any linear functional $$L:mathbb {R}[x_1,dots ,x_n]rightarrow mathbb {R}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>:</mml:mo> <mml:mi>R</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> is a K -moment functional if and only if it has a continuous extension to some $$overline{L}:mathbb {R}[x_1,dots ,x_n]+mathbb {R}[g]rightarrow mathbb {R}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>:</mml:mo> <mml:mi>R</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>R</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>g</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> such that $$tilde{L}:mathbb {R}[t]rightarrow mathbb {R}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>:</mml:mo> <mml:mi>R</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>t</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> defined by $$tilde{L}(t^d):= overline{L}(g^d)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>g</mml:mi> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for all $$din mathbb {N}_0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> is a [0, 1]-moment functional (Hausdorff moment problem). Additionally, there exists a continuous function $$f:[0,1]rightarrow K$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> <mml:mo>→</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> independent on L such that the representing measure $$tilde{mu }$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mover> <mml:mi>μ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> of $$tilde{L}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mover> <mml:mi>L</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> provides the representing measure $$tilde{mu }circ f^{-1}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mover> <mml:mi>μ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>∘</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> of L . We also show that every moment functional $$L:mathcal {V}rightarrow mathbb {R}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>:</mml:mo> <mml:mi>V</mml:mi> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> is represented by $$lambda circ f^{-1}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∘</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> for some measurable function $$f:[0,1]rightarrow mathbb {R}^n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> where $$lambda $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>λ</mml:mi> </mml:math> is the Lebesgue measure on [0, 1]." @default.
• W3044062023 created "2020-07-29" @default.
• W3044062023 creator A5010485476 @default.
• W3044062023 date "2022-12-15" @default.
• W3044062023 modified "2023-10-18" @default.
• W3044062023 title "Transformations of Moment Functionals" @default.
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