A New Discrete Representation of Linear Functionals

Abstract

Given a linear functional $\mathscr {U}$ in the linear space $\mathbb {P}$ of polynomials in one variable with complex coefficients, under certain conditions, we give a new representation of a $\mathscr {U}$ by a discrete measure. Finally, as an application for arbitrary $q\in \mathbb {C}$ with $|q|<1,$ we can represent the linear functionals $\delta '$ and $2q\delta -\delta ''$ by means of discrete measures with $\hbox {supp}=\{q^{-k},\;k\ge 0\},$ with $\delta$ the Dirac functional defined by $\langle \delta ,p\rangle :=\;p(0),\; p\in \mathbb {P}$.