Positivity and positivity-definiteness for Cauchy powers of linear functionals on the linear space of polynomials (Q6552233)

Let \(\mathfrak{U}\) and \(\mathfrak{V}\) be normalized linear functionals on the space of real polynomials. The main goal of this study is to determine the foundational outcome:\N\[\N\left. \begin{array}{c} \mathfrak{VU}\text{ is positive-definite} \\\N\text{and} \\\N\mathfrak{VU}^{-1}\text{ is positive} \end{array} \right\} \Longrightarrow \mathfrak{V}\text{ is positive-definite}\N\]\NThe author begins by extending the notion of the integer power of a linear functional to achieve the secondary goal, which is to introduce the concept of the index of positivity for a normalized linear functional on real polynomials, including the Dirac mass at any real point and some linear functionals with semi-classical character.