Positivity and Positivity-Definiteness for Cauchy Powers of Linear Functionals on the Linear Space of Polynomials

In this paper, an exploration is undertaken into positivity and positivity-definiteness within the Cauchy product and self-product, which encompass normalized linear functionals applied to the space of real polynomials. We reveal that for two normalized linear functionals, \(\mathscr {U}\) and \(\mathscr {V}\), the positivity-definiteness of \(\mathscr {V}\mathscr {U}\) and the positivity of \(\mathscr {V}\mathscr {U}^{-1}\) imply the positive-definiteness of \(\mathscr {V}\). Additionally, if \(\mathscr {U}^2\) is positive-definite (resp. positive), and \(\mathscr {V}^2\) is positive, then \(\mathscr {U}\mathscr {V}\) is positive-definite (resp. positive). The extension of the integer Cauchy power to the real powers of a linear functional introduces the concept of the index of positivity for linear functionals. We establish some properties of the index map. Finally, we determine the index of positivity for various linear functionals, including the Dirac mass at any real point and some linear functionals with semi-classical character.