## 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{eqnarray*} (f,g)_S=(f(c_0),f(c_1),\ldots,f(c_{N-1})){\bf A}\left( \begin{array}{c} g(c_0) \ g(c_1) \ \vdots \ g(c_{N-1}) \end{array} \right)+ \langle u,f^{(N)}g^{(N)} \rangle, \end{eqnarray*} where $u$ is a quasi-definite (or regular) linear functional on the linear space ${\bf P}$ of real polynomials, $c_0,c_1,\ldots,c_{N-1}$ are distinct real numbers, $N$ is a positive integer number, and ${\bf A}$ 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.

33C45, 42C05.

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