an:01513212
Zbl 0980.42017
Alfaro, M.; P??rez, T. E.; Pi??ar, M. A.; Rezola, M. L.
Sobolev orthogonal polynomials: The discrete-continuous case
EN
Methods Appl. Anal. 6, No. 4, 593-616 (1999).
00068250
1999
j
42C05 33C45
Sobolev orthogonal polynomials; classical orthogonal polynomials; Sobolev bilinear form; second order differential equation; recurrence relation
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.
Andrei Mart??nez Finkelshtein (Almeria)