Polynomial Interpolation on Quasi-Equidistributed Nodes on the Unit Disk

Abstract

Polynomial interpolation for the analytic function on the unit disk in the complex plane is considered. The polynomial interpolation on a quasi-equidistributed node set which is a union of several rotations around the origin for an equidistributed node set on the unit circle is investigated. A scheme for generating a sequence of quasi-equidistributed node sets for polynomial interpolation on the unit disk is proposed, in which each set is contained in its successor and the average increased rate of number of nodes at each step is less than 2, in fact, may be arbitrarily close to 1. An algorithm, based on the FFT technique, is also proposed for constructing the sequence of interpolation polynomials successively on the sequence of node sets generated by the scheme. This algorithm requires $\frac{1}{2}n\log _2 n + O(n)$ complex multiplications for n nodes. Our interpolation process is as stable as the ordinary process on equidistributed node sets, since the Lebesgue constant of the corresponding interpolatio...