A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomi...

ABSTRACT. Free Meixner states are a class of functionals on non-commutative polynomials introduced in [Ans06]. They are characterized by a resolvent-type form for the generating function of their orthogonal polynomials, by a recursion relation for those polynomials, or by a second-order non-commutative differential equation satisfied by their free cumulant functional. In this paper, we construc...

in this study, the problem of determining an optimal trajectory of a nonlinear injection into orbit problem with minimum time was investigated. the method was based on orthogonal polynomial approximation. this method consists of reducing the optimal control problem to a system of algebraic equations by expanding the state and control vector as chebyshev or legendre polynomials with undetermined...

A pair of quasi-de nite linear functionals fu0; u1g on the set of polynomials is called a coherent pair if their corresponding sequences of monic orthogonal polynomials fPng and fTng satisfy a relation Tn = P 0 n+1 n+ 1 nP 0 n n ; n 1; with n non-zero constants. We prove that if fu0; u1g is a coherent pair, then at least one of the functionals has to be classical, i.e. Hermite, Laguerre, Jacobi...