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Representation of measures with polynomial denseness in $ \mathbf{L}_{p}\, (\mathbb{R}, d\mu)$, $ 0<p<\infty$, and its application to determinate moment problems

Author: Andrew G. Bakan
Journal: Proc. Amer. Math. Soc. 136 (2008), 3579-3589
MSC (2000): Primary 46E30, 41A10; Secondary 44A60, 41A65
Published electronically: June 4, 2008
MathSciNet review: 2415042
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Abstract: It has been proved that algebraic polynomials $ \mathcal{P}$ are dense in the space $ L^{p}({\mathbb{R}},d\mu)$, $ p\in(0, \infty)$, iff the measure $ \mu$ is representable as $ d\mu=w^p\, d\nu$ with a finite non-negative Borel measure $ \nu$ and an upper semi-continuous function $ w:\mathbb{R}\to\mathbb{R}^+:\,=[0,\infty)$ such that $ \mathcal{P}$ is a dense subset of the space $ C^0_w :\,= \{f\in C(\mathbb{R}) : w(x)f (x)\to 0 \,$   as$ \, \vert x\vert\to\infty \}$ 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)d\mu=w^2 d\nu$ ( $ (1+x)d\mu=w^2 d\nu$) 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}},d\mu)$ has also been examined.

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Andrew G. Bakan
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska Street 3, Kyiv 01601, Ukraine

Keywords: Spaces of measurable functions, approximation by polynomials, moment problems
Received by editor(s): August 21, 2007
Published electronically: June 4, 2008
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society