Volume 9, pp. 56-64, 1999.

Sobolev orthogonal polynomials: interpolation and approximation

Esther M. García-Caballero, Teresa E. Pérez, and Miguel A. Piñar

Abstract

In this paper, we study orthogonal polynomials with respect to the bilinear form $$\begin{array}{r}(f,g{)}_S=(f(c_0),f(c_1),\dots ,f(c_{N-1}))\mathbf{A}\left(\begin{array}{c}g(c_0)g(c_1)⋮g(c_{N-1})\end{array}\right)+⟨u,f^{(N)}{g}^{(N)}⟩,\end{array}$$ where $u$ is a quasi-definite (or regular) linear functional on the linear space $\mathbf{P}$ of real polynomials, $c_0,c_1,\dots ,c_{N-1}$ are distinct real numbers, $N$ is a positive integer number, and $\mathbf{A}$ is a real $N\times N$ matrix such that each of its principal submatrices are nonsingular. We show a connection between these non-standard orthogonal polynomials and some standard problems in the theory of interpolation and approximation.

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Key words

Sobolev orthogonal polynomials, classical orthogonal polynomials, interpolation, approximation.

AMS subject classifications

33C45, 42C05.