Transformations of Moment Functionals

• Published in: Integral equations and operator theory 2023-03, Vol.95 (1), Article 2
• Main Author: di Dio, Philipp J.
• Format: Article
• Language: English
• Subjects: Analysis; Continuity (mathematics); Mathematics; Mathematics and Statistics
• ISSN: 0378-620X; 1420-8989
• DOI: 10.1007/s00020-022-02722-3

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 K ⊂ R n there exists a measurable function g : K → [ 0 , 1 ] such that any linear functional L : R [ x 1 , ⋯ , x n ] → R is a K -moment functional if and only if it has a continuous extension to some L ¯ : R [ x 1 , ⋯ , x n ] + R [ g ] → R such that L ~ : R [ t ] → R defined by L ~ ( t d ) : = L ¯ ( g d ) for all d ∈ N 0 is a [0, 1]-moment functional (Hausdorff moment problem). Additionally, there exists a continuous function f : [ 0 , 1 ] → K independent on L such that the representing measure μ ~ of L ~ provides the representing measure μ ~ ∘ f - 1 of L . We also show that every moment functional L : V → R is represented by λ ∘ f - 1 for some measurable function f : [ 0 , 1 ] → R n where λ is the Lebesgue measure on [0, 1].