Positivity and positivity-definiteness for Cauchy powers of linear functionals on the linear space of polynomials. (English) Zbl 07861936
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:
\[
\left. \begin{array}{c} \mathfrak{VU}\text{ is positive-definite} \\
\text{and} \\
\mathfrak{VU}^{-1}\text{ is positive} \end{array} \right\} \Longrightarrow \mathfrak{V}\text{ is positive-definite}
\]
The 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.
Reviewer: Elhadj Dahia (Bou Saâda)
MSC:
| 46G25 | (Spaces of) multilinear mappings, polynomials |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
Keywords:
positive linear functional; positive-definite linear functional; weakly regular linear functional; regular linear functional; orthogonal polynomial sequence; Cauchy-exponential of linear functional; Cauchy-power and self-power of linear functional; index of positivity of linear functionalReferences:
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