Sobolev orthogonal polynomials: The discrete-continuous case. (English) Zbl 0980.42017

Alfaro, M.; Pérez, T. E.; Piñar, M. A.; Rezola, M. L.

Methods Appl. Anal. 6, No. 4, 593-616 (1999).

If a sequence of polynomials is orthogonal with respect to a bilinear form involving derivatives, these are known as Sobolev orthogonal polynomials. In this paper, a particular case of the bilinear form is considered, called the discrete-continuous one, such as that it involves up to \(N \in \mathbb N\) derivatives of the functions, but the first \(N-1\) appear evaluated only at a fixed point \(c \in \mathbb R\).
The authors accomplish a thorough study of the algebraic and differential properties of the corresponding Sobolev orthogonal polynomials and of their connection with the standard orthogonal polynomials. In particular, a new characterization of classical polynomials (as the only orthogonal polynomials that for some \(N \in \mathbb N\) have an \(N\)-th primitive satisfying a three-term recurrence relation) is given.

Reviewer: Andrei Martínez Finkelshtein (Almeria)

MSC:

42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

Sobolev orthogonal polynomials; classical orthogonal polynomials; Sobolev bilinear form; second order differential equation; recurrence relation