Rational moment problems for compact sets
The author treats the following `rational' moment problem: given distinct real numbers \(\lambda_ 1,\dots,\lambda_ p\) and real numbers \(c_ 0\), \(c_ j^{(i)}\) \((j\geq 1,\;1\leq i\leq p)\) and a non- empty compact subset \({\mathbf K}\) of the real line, find a non-negative Borel measure \(\mu\) supported on \({\mathbf K}\) with \[ c_ 0= \int_{{\mathbf K}} d\mu(t),\;c_ j^{(i)}= \int_{{\mathbf K}}(t- \lambda_ i)^{-j} d\mu(t). \] By looking at the truncated moment problem (restrict \(j: 1\leq j\leq 2n_ i\), \(2n_ i\) positive even integers), the author derives necessary and sufficient conditions for the solvability of the rational moment problem in terms of the positivity of certain linear functionals on the space of rational functions of the form \[ P(t)/\prod_{i=1}^ p (t-\lambda_ i)^{2n_ i} \] with \(P(t)\) a polynomial of degree \(\leq 2(n_ 1+\cdots+ n_ p)\). Also the case with countably many \(\lambda\)'s is treated. Moreover the finite and countably infinite case of the problem with a pole at infinity (replace \(c_ 0\), by \(c_ j^{(0)}\) and the conditions for those moments are \(c_ j^{(0)}=\int_{{\mathbf K}} t^ j d\mu(t)\)) are solved. The paper concludes with the construction of a positive Borel measure \(\mu\) with finite total mass and support equal to the closed interval \([0,2]\) such that \((t- \lambda_ i)^{-j}\), \(t^ j\) \((i\geq 0, j\geq 0)\) belong to \(L^ 1(\mu)\), where the \(\lambda\)'s satisfy \(0< \lambda_ 1<\lambda_ 2<\cdots< \lambda_ 0< 1\) and \(\lambda_ n\to \infty\) as \(n\to \infty\). A nicely written, interesting paper.
scientific article; zbMATH DE number 645058