Page 101 - Textos de Matemática Vol. 34

TOPOLOGICAL ASPECTS IN THE THEORY OF ORTHOGONAL POLYNOMIALS AND AN INVERSE PROBLEM
JOSE´ CARLOS PETRONILHO
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
Abstract. A review on topological aspects in the theory of orthogonal poly- nomials is presented. As an application, we present an alternative proof to a recent result concerning linearly related sequences of orthogonal polynomials and their functionals (J. Math. Anal. Appl. 287 (2003), no. 1, 307–319). We do not claim originality in the results presented in the paper up to the last section, unless, perhaps, in some aspects of their presentation.
1. Introduction
Orthogonal polynomials can be studied from many different points of view, ac- cording to the motivations of the authors dealing with the subject. In this paper we follow the approach developed by Pascal Maroni, so that orthogonality of a sequence of polynomials is defined with respect to a moment linear functional be- longing to the topological dual space P′ of the space P, where, by definition, P is C[x] carried with a structure of a locally convex space with the topology of the hyper-strict inductive limit defined by the sequence of spaces {Pn}n≥0, Pn being the finite dimensional normed space of the polynomials of degree ≤ n. The main fact in the basis of this approach is the set equality
(1.1) P′ = P∗ ,
where P∗ denotes the algebraic dual of P. In fact, (1.1) simply tells us that every linear functional defined in P is continuous. This property never holds in an infinite dimensional normed space, since in such a space one can always ensure the existence of a linear functional which is not continuous (a well known fact which can be proved by using Zorn’s Lemma), hence P is not normable (in fact, it is not metrizable).
2000 Mathematics Subject Classification. 42C05, 33C47, 46A13.
Key words and phrases. Orthogonal polynomials, moment linear functionals, locally convex spaces, inverse problems, Sobolev orthogonal polynomials.
The work was supported by CMUC (Centro de Matem´atica da Universidade de Coimbra) and by Acc¸˜ao Integrada Luso-Espanhola E-6/03.
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