General Sobolev orthogonal polynomials (Q1919576)

The authors derive properties of polynomials orthogonal with respect to the inner product \[ (f,g)=\langle u,fg\rangle+ \sum^N_{m=1} \lambda_m\langle u,f^{(m)} g^{(m)}\rangle, \] where the constant \(\lambda_m\) are nonnegative and \(u\) is a semiclassical, positive definite linear functional, so called Sobolev orthogonal polynomials. The connection between these polynomials and the semiclassical orthogonal polynomials with respect to the functional \(u\) is investigagted. Furthermore, algebraic and differential properties are obtained. Some examples illustrating the general considerations are added.