On the computation of sets of points with low Lebesgue constant on the unit disk

In this paper of numerical nature, we test the Lebesgue constant of several available point sets on the disk and propose new ones that enjoy low Lebesgue constant. Furthermore we extend some results in Cuyt (2012), analyzing the case of Bos arrays whose radii are nonnegative Gauss–Gegenbauer–Lobatto nodes with exponent, noticing that the optimal still allows to achieve point sets on with low Lebesgue constant for degrees. Next we introduce an algorithm that through optimization determines point sets with the best known Lebesgue constants for. Finally, we determine theoretically a point set with the best Lebesgue constant for the case

Lagrange interpolation at real projections of Leja sequences for the unit disk

We show that the Lebesgue constant of the real projection of Leja sequences for the unit disk grows like a polynomial. The main application is the first construction of explicit multivariate interpolation points in $[-1,1]^N$ whose Lebesgue constant also grows like a polynomial.Comment: 12 pages, 2 figure

The Penalized Lebesgue Constant for Surface Spline Interpolation

Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying density has not been investigated with the same intensity, and is far less well understood. In this article we consider the stability of surface spline interpolation (a popular type of RBF interpolation) for data with nonuniform arrangements. Using techniques similar to those recently employed by Hangelbroek, Narcowich and Ward to demonstrate the stability of interpolation from quasi-uniform data on manifolds, we show that surface spline interpolation on R^d is stable, but in a stronger, local sense. We also obtain pointwise estimates showing that the Lagrange function decays very rapidly, and at a rate determined by the local spacing of datasites. These results, in conjunction with a Lebesgue lemma, show that surface spline interpolation enjoys the same rates of convergence as those of the local approximation schemes recently developed by DeVore and Ron.Comment: 20 pages; corrected typos; to appear in Proc. Amer. Math. So

The Lebesgue Constant for the Periodic Franklin System

We identify the torus with the unit interval $[0,1)$ and let $n,\nu\in\mathbb{N}$, $1\leq \nu\leq n-1$ and $N:=n+\nu$. Then we define the (partially equally spaced) knots t_{j}=\{[c]{ll}% \frac{j}{2n}, & \text{for}j=0,...,2\nu, \frac{j-\nu}{n}, & \text{for}j=2\nu+1,...,N-1.] Furthermore, given $n,\nu$ we let $V_{n,\nu}$ be the space of piecewise linear continuous functions on the torus with knots $\{t_j:0\leq j\leq N-1\}$. Finally, let $P_{n,\nu}$ be the orthogonal projection operator of $L^{2}([0,1))$ onto $V_{n,\nu}.$ The main result is \[\lim_{n\rightarrow\infty,\nu=1}\|P_{n,\nu}:L^\infty\rightarrow L^\infty\|=\sup_{n\in\mathbb{N},0\leq \nu\leq n}\|P_{n,\nu}:L^\infty\rightarrow L^\infty\|=2+\frac{33-18\sqrt{3}}{13}. This shows in particular that the Lebesgue constant of the classical Franklin orthonormal system on the torus is $2+\frac{33-18\sqrt{3}}{13}$.Comment: Mathematica Notebook for creating Table 1 on page 21 is attache

Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant

The purpose of this paper is to establish that for any compact, connected C^{\infty} Riemannian manifold there exists a robust family of kernels of increasing smoothness that are well suited for interpolation. They generate Lagrange functions that are uniformly bounded and decay away from their center at an exponential rate. An immediate corollary is that the corresponding Lebesgue constant will be uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The analysis needed for these results was inspired by some fundamental work of Matveev where the Sobolev decay of Lagrange functions associated with certain kernels on \Omega \subset R^d was obtained. With a bit more work, one establishes the following: Lebesgue constants associated with surface splines and Sobolev splines are uniformly bounded on R^d provided the data sites \Xi are quasi-uniformly distributed. The non-Euclidean case is more involved as the geometry of the underlying surface comes into play. In addition to establishing bounded Lebesgue constants in this setting, a "zeros lemma" for compact Riemannian manifolds is established.Comment: 33 pages, 2 figures, new title, accepted for publication in SIAM J. on Math. Ana