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Representation of measures with polynomial denseness in , , and its application to determinate moment problems
Authors:Andrew G. Bakan
Affiliation:Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska Street 3, Kyiv 01601, Ukraine
Abstract:
It has been proved that algebraic polynomials $ mathcal{P}$ are dense in the space $ L^{p}({mathbb{R}},dmu)$, $ pin(0, infty)$, iff the measure $ mu$ is representable as $ dmu=w^p, dnu$ with a finite non-negative Borel measure $ nu$ and an upper semi-continuous function $ w:mathbb{R}tomathbb{R}^+:,=[0,infty)$ such that $ mathcal{P}$ is a dense subset of the space $ C^0_w :,= {fin C(mathbb{R}) : w(x)f (x)to 0 ,$   as$ , vert xverttoinfty }$ equipped with the seminorm $ Vert f Vert _{w}:= sup_{x in{mathbb{R}}} w(x)vert f(x)vert$. The similar representation $ (1+x^2)dmu=w^2 dnu$ ( $ (1+x)dmu=w^2 dnu$) with the same $ nu$ and $ w$ ( $ w(x)=0, x < 0$, and $ mathcal{P}$ is also a dense

subset of $ {C^0_{sqrt{x},cdot, w}}$) corresponds to all those measures (supported by $ mathbb{R}^+$) that are uniquely determined by their moments on $ mathbb{R}$ ( $ mathbb{R}^+$). The proof is based on de Branges' theorem (1959) on weighted polynomial approximation. A more general question on polynomial denseness in a separable Fréchet space in the sense of Banach $ L^Phi({mathbb{R}},dmu)$ has also been examined.

Keywords:Spaces of measurable functions   approximation by polynomials   moment problems
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