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Trends and Applications in Constructive Approximation (Eds.) M.G. de Bruin, D.H. Mache & J. Szabados International Series of Numerical Mathematics, Vol. 151, 61–70 c 2005 Birkh¨ auser Verlag Basel/Switzerland
(0, 2) P´al-type Interpolation: A General Method for Regularity Marcel G. de Bruin and Detlef H. Mache Abstract. The methods of proof of regularity for interpolation problems often are dependent on the problem at hand. In case of given pairs of node generating polynomials the method of deriving an ordinary differential equation for the interpolating polynomial or that of exploiting the specific form of the node generator have mainly been used up to now. Recently another method was used in the case of P´ al-type interpolation where ‘only’ one of the node generators is fixed in advance: a ‘general’ method of deriving a companion generator that leads to a regular interpolation problem. Using (0, 2) P´ al-type interpolation, it is shown that each of the methods has its merits and for sake of simplicity we will restrict ourselves to the case that the nodes are the zeros of pairs of polynomials of the following form: {p(z)q(z), p(z)} with p, q co-prime and both having simple zeros. Mathematics Subject Classification (2000). 41A05. Keywords. P´ al type interpolation, regularity.
1. Introduction The study of Hermite-Birkhoff interpolation is a well-known subject (cf. the excellent book [2]). Recently the regularity of some interpolation problems on nonuniformly distributed nodes on the unit circle has been studied. Along with the continuing interest in interpolation in general, a number of papers on P´ al-type interpolation have appeared, cf. [3], [4], [6]. In this paper the attention will be focused on so-called (0, 2) P´ al-type interpolation problems on the pair of node generators {p(z)q(z), p(z)}: – given two co-prime polynomials p(z) resp. q(z), with simple zeros {zi }ni=1 ∈ C res