Manuel Alfaro

Orthogonal polynomials on Sobolev spaces: old and new directions

Abstract During the last years, orthogonal polynomials on Sobolev spaces have attracted considerable attention. Algebraic properties, distribution of their zeros and Fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case. In this paper we present an introductory survey about the subject. The origin of the problems and their development show the interest and the promising future of this field.

On linearly related orthogonal polynomials and their functionals

Let {Pn} be a sequence of polynomials orthogonal with respect a linear functional u and {Qn} a sequence of polynomials defined by Pn(x) + snPn−1(x) = Qn(x) + tnQn−1(x). We find necessary and sufficient conditions in order to {Qn} be a sequence of polynomials orthogonal with respect to a linear functional v. Furthermore we prove that the relation between these linear functionals is (x −˜ a)u = λ(x − a)v. Even more, if u and v are linked in this way we get that {Pn} and {Qn} satisfy a formula as above.

Orthogonality of the Jacobi polynomials with negative integer parameters

It is well known that the Jacobi polynomials Pn(α,β)(x) are orthogonal with respect to a quasi-definite linear functional whenever α, β, and α + β + 1 are not negative integer numbers. Recently, Sobolev orthogonality for these polynomials has been obtained for α a negative integer and β not a negative integer and also for the case α = β negative integer numbers.In this paper, we give a Sobolev orthogonality for the Jacobi polynomials in the remainder cases.

Some properties of zeros of Sobolev-type orthogonal polynomials

Abstract For polynomials orthogonal with respect to a discrete Sobolev product, we prove that, for each n , Q n has at least n − m zeros on the convex hull of the support of the measure, where m denotes the number of terms in the discrete part. Interlacing properties of zeros are also described.

Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs

Abstract Let Qn be the polynomials orthogonal with respect to the Sobolev inner product (f,g) S =∫ fg d μ 0 +∫ f′g′ d μ 1 , being (μ0,μ1) a coherent pair where one of the measures is the Hermite measure. The outer relative asymptotics for Qn with respect to Hermite polynomials are found. On the other hand, we consider the Sobolev scaled polynomials and we obtain the Plancherel–Rotach asymptotics for those as well as a consequence about their zeros.

Asymptotics for a generalization of Hermite polynomials

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler{Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.

On linearly related orthogonal polynomials in several variables

Let {ℙn}n≥0

Orthogonal polynomials generated by a linear structure relation: Inverse problem

\{\mathbb{P}_{n}\}_{n\ge 0}

Semiorthogonal functions and orthogonal polynomials on the unit circle

and {ℚn}n≥0

A Cubic Decomposition of Sequences of Orthogonal Polynomials on the Unit Circle

\{\mathbb{Q}_{n}\}_{n\ge 0}