## Bivariate interpolation at Xu points: results, extensions and applications

Len Bos, Marco Caliari, Stefano De Marchi, and Marco Vianello

### Abstract

In a recent paper, Y. Xu proposed a set of Chebyshev-like points for polynomial interpolation on the square $[-1,1]^2$. We have recently proved that the Lebesgue constant of these points grows like $\log^2$ of the degree (as with the best known points for the square), and we have implemented an accurate version of their Lagrange interpolation formula at linear cost. Here we construct non-polynomial Xu-like interpolation formulas on bivariate compact domains with various geometries, by means of composition with suitable smooth transformations. Moreover, we show applications of Xu-like interpolation to the compression of surfaces given as large scattered data sets.

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### Key words

bivariate polynomial interpolation, Xu points, Lebesgue constant, domains transformations, generalized rectangles, generalized sectors, large scattered data sets, surface compression

65D05

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