On orthogonal polynomials for certain non-definite linear functionals
نویسنده
چکیده
We consider the non-definite linear functionals Ln[f ] = ∫ IR w(x)f (n)(x) dx and prove the nonexistence of orthogonal polynomials, with all zeros real, in several cases. The proofs are based on the connection with moment preserving spline approximation.
منابع مشابه
On orthogonal polynomials for certain nonde nite linear functionals
We consider the non-de nite linear functionals Ln[f] = ∫ R w(x)f (x) dx and prove the nonexistence of orthogonal polynomials, with all zeros real, in several cases. The proofs are based on the connection with moment preserving spline approximation. c © 1998 Elsevier Science B.V. All rights reserved.
متن کاملPolynomial perturbations of hermitian linear functionals and difference equations
This paper is devoted to the study of general (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree. The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a ...
متن کاملMatrix orthogonal polynomials whose derivatives are also orthogonal
In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general...
متن کاملDarboux transformation and perturbation of linear functionals
Let L be a quasi-definite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: xL, L + Cδ(x) and x L + Cδ(x) where δ(x) denotes the linear functional (δ(x))(xk) = δk,0, and δk,0 is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L....
متن کاملq-Coherent pairs and q-orthogonal polynomials
In this paper we introduce the concept of q coherent pair of linear functionals. We prove that if ðu0; u1Þ is a q coherent pair of linear functionals, then at least one of them has to be a q classical linear functional. Moreover, we present the classification of all q coherent pairs of positive definite linear functionals when u0 or u1 is either the little q Jacobi linear functional or the litt...
متن کامل