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

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 $C^0_w : = \{f\in C(\mathbb {R}) : w(x)f (x)\to 0 \mbox {as} |x|\to \infty \}$ equipped with the seminorm $\| f \|_{w}:= \sup _{x \in {\mathbb {R}}} w(x)|f(x)|$. 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.