Portugaliae Mathematica   EMIS ELibM Electronic Journals PORTUGALIAE
Vol. 56, No. 1, pp. 81-113 (1999)

Previous Article

Next Article

Contents of this Issue

Other Issues

ELibM Journals

ELibM Home



Orthogonal Polynomials and Quadratic Transformations

Francisco Marcellán and José Petronilho

Depto. de Matemáticas, Escuela Politécnica Superior, Univ. Carlos III de Madrid,
Butarque 15, 28911 Leganés, Madrid - SPAIN
E-mail: pacomarc@ing.uc3m.es
Depto. de Matemática, Faculdade de Ciências e Tecnologia, Univ. Coimbra,
Apartado 3008, 3000 Coimbra - PORTUGAL
E-mail: josep@mat.uc.pt

Abstract: Starting from a sequence $\{P_n\}_{n\geq 0}$ of monic polynomials orthogonal with respect to a linear functional ${\bf u}$, we find a linear functional ${\bf v}$ such that $\{Q_n\}_{\geq 0}$, with either $Q_{2n}(x)=P_n(T(x))$ or $Q_{2n+1}(x)=(x-a)\,P_n(T(x))$ where $T$ is a monic quadratic polynomial and $a\in\C$, is a sequence of monic orthogonal polynomials with respect to ${\bf v}$. In particular, we discuss the case when ${\bf u}$ and ${\bf v}$ are both positive definite linear functionals. Thus, we obtain a solution for an inverse problem which is a converse, for quadratic mappings, of one analyzed in [11].

Keywords: Orthogonal polynomials; recurrence coefficients; polynomial mappings; Stieltjes functions.

Classification (MSC2000): 42C05.

Full text of the article:

Electronic version published on: 31 Jan 2003. This page was last modified: 27 Nov 2007.

© 1999 Sociedade Portuguesa de Matemática
© 1999–2007 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition