On inverse problem leading to second-order linear functionals

A linear functional LL is said to be positive-definite if and only if 〈L,p2〉>0〈L,p2〉>0, for all non-zero polynomials with real coefficients pp. In this paper, we provide a new construction process of a positive-definite linear functional from positive-definite linear functional data. Indeed, for any non-zero real ϵϵ and any positive-definite linear functional LL, we show that the linear functional LϵLϵ satisfying Lϵ−ϵLϵ′=L is also positive-definite. This process allows us to construct a second-order positive-definite linear functional from a semiclassical positive-definite linear functional. Finally, we give an illustrative example.