We give a simple, geometric and explicit construction of bivariate interpolation at certain points in a square (called Padua points), giving compact formulas for their fundamental Lagrange polynomials. We show that the associated norms of the interpolation operator, i.e., the Lebesgue constants, have minimal order of growth of O((log n) 2 ). To the best of our knowledge this is the first complete, explicit example of near optimal bivariate interpolation points.

We discuss and compare two greedy algorithms, that compute discrete versions of Fekete-like points for multivariate compact sets by basic tools of numerical linear algebra. The first gives the so-called "Approximate Fekete Points" by QR factorization with column pivoting of Vandermonde-like matrices. The second computes Discrete Leja Points by LU factorization with row pivoting. Moreover, we study the asymptotic distribution of such points when they are extracted from Weakly Admissible Meshes.2000 AMS subject classification: 41A10, 41A63, 65D05.

The goal of this paper is to construct data-independent optimal point sets for interpolation by radial basis functions. The interpolation points are chosen to be uniformly good for all functions from the associated native Hilbert space. To this end we collect various results on the power function, which we use to show that good interpolation points are always uniformly distributed in a certain sense. We also prove convergence of two different greedy algorithms for the construction of near-optimal sets which lead to stable interpolation. Finally, we provide several examples.

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It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered, the L 2 or L ∞ norms of interpolants can be bounded above by discrete 2 and ∞ norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample size and the fill distance.