Padua points and fake nodes for polynomial approximation: old, new and open problems

Abstract

Padua points, discovered in 2005 at the University of Padua, are the first set of points on the square $[-1,1{]}^2$ that are explicitly known, unisolvent for total degree polynomial interpolation and with Lebesgue constant increasing like ${\log }^2\left(n\right)$ of the degree. One of the key features of the Padua points is that they lie on a particular Lissajous curve. Other important properties of Padua points are

1. in two dimensions, Padua points are a WAM for interpolation and for extracting approximate Fekete points and discrete Leja sequences.
2. in three dimensions, Padua points can be used for constructing tensor product WAMs on different compacts.

Unfortunately, their extension to higher dimensions is still the biggest open problem. 

The concept of mapped bases has been widely studied (cf. e.g. [35] and references therein), which turns out to be equivalent to map the interpolating nodes and then construct the approximant in the classical form without the need of resampling. The mapping technique is general, in the sense that works with any basis and can be applied to continuous, piecewise or discontinuous functions or even images.
All the proposed methods show convergence to the interpolant provided that the function is resampled at the mapped nodes. In applications, this is often physically unfeasible. An effective method for interpolating via mapped bases in the multivariate setting, referred as Fake Nodes Approach (FNA), has been presented in [37]. In this paper, some interesting connection of the FNA with Padua points and families of relatives nodes, that can be used as fake nodes for multivariate approximation, are presented and we conclude with some open problems.

Keywords

• Padua points
• Lissajous curves and points
• mapped polynomial basis.

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