Page 102 - Textos de Matemática Vol. 34

92 JOSE´ CARLOS PETRONILHO
Nevertheless, the topology (of the hyper-strict inductive limit) introduced in P yields the possibility of giving a meaning to the convergence of any sequence of functionals in P∗, in the sense of the weak topology in P′. In particular, a Fourier- type expansion of any given linear functional as a linear combination (finite or not) of the elements of any dual basis corresponding to any simple set in C[x] is always possible. This fact enables us to deal directly in the dual space P′ —by choosing some appropriate dual basis when the corresponding simple sets are sequences of orthogonal polynomials or not— instead of work in the space P of the polynomials. Thus, the use of continued fractions and recurrence relations —which are the main classical tools when working with algebraic properties of orthogonal sequences in P— is replaced by the use of dual basis in P′ of appropriate chosen simple sets in P. This is the key observation which is in the basis of the theory developed by Pascal Maroni. In fact, the application of these ideas produces a natural way for the study of the algebraic properties of sequences of orthogonal polynomials. Further, in the so-called positive definite case, often it also leads to the necessary understanding of the problem in consideration in order to get the analytic properties (in particular, the orthogonality measure) of the polynomials.
The paper is organized as follows. In sections 2–7 we show how the theory of locally convex spaces can be applied to provide the space of the polynomials and its dual with adequate topological tools. Most of the results in these sections are contained in the papers [11, 13, 15] by Maroni (although in these papers some of the proofs were omitted or are different from the ones we present here), as well as in the standard textbooks by Szeg¨o [20], Chihara [4] and Freud [5]. In the last section we consider an inverse problem in the theory of orthogonal polynomials concerning linearly related sequences of orthogonal polynomials and their linear functionals. This problem has been recently solved by M. Alfaro, F. Marcell´an, A. Pen˜a and M. L. Rezola [1]. We provide an alternative proof based on the results presented in the previous sections. Such kind of problems has received much attention in the last years, specially in the study of the so-called Sobolev-type orthogonal polynomials.
2. The spaces P and P′
The topology in which we are interested for the linear space of the polynomials belongs to the “ inductive limit topologies”. These topologies are defined in a sub- class of the class of the topological vector spaces (t.v.s.), namely, the locally convex spaces (l.c.s.). All the facts needed in the paper concerning the theory of these spaces are contained, e.g., in the books by F. Treves [21] or M. Reed and B. Simon [18] (see also [2, 6, 19]).
With the usual operations of addition and scalar multiplication, C[x] is a linear space which will be denoted by P. Pn will denote the linear subspace of P of polynomials with degree less than or equal to n. Since in a finite dimensional