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

Padua points, discovered in 2005 at the University of Padua, are first set points on square [−1,1]2[−1,1]2 that explicitly known, unisolvent for total degree polynomial interpolation and with Lebesgue constant increasing like log2(n)log2⁡(n) degree. One key features is they lie a particular Lissajous curve. Other important properties arein two dimensions, WAM extracting approximate Fekete discrete Leja sequences.in three can be used constructing tensor product WAMs different compacts.Unfortunately, their extension to higher dimensions still biggest open problem. The concept mapped bases has been widely studied (cf. e.g. [35] references therein), which turns out equivalent map interpolating nodes then construct approximant classical form without need resampling. mapping technique general, sense works any basis applied continuous, piecewise or discontinuous functions even images. All proposed methods show convergence interpolant provided function resampled nodes. In applications, this often physically unfeasible. An effective method via multivariate setting, referred as Fake Nodes Approach (FNA), presented [37]. paper, some interesting connection FNA families relatives nodes, fake approximation, we conclude problems.