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

Optimal sampling patterns for Zernike polynomials

A pattern of interpolation nodes on the disk is studied, for which the inter- polation problem is theoretically unisolvent, and which renders a minimal numerical condition for the collocation matrix when the standard basis of Zernike polynomials is used. It is shown that these nodes have an excellent performance also from several alternative points of view, providing a numer- ically stable surface reconstruction, starting from both the elevation and the slope data. Sampling at these nodes allows for a more precise recovery of the coefficients in the Zernike expansion of a wavefront or of an optical surface. Keywords: Interpolation Numerical condition Zernike polynomials Lebesgue constant

Size of orthogonal sets of exponentials for the disk

Suppose \Lambda \subseteq \RR^2 has the property that any two exponentials with frequency from $\Lambda$ are orthogonal in the space $L^2(D)$, where D \subseteq \RR^2 is the unit disk. Such sets $\Lambda$ are known to be finite but it is not known if their size is uniformly bounded. We show that if there are two elements of $\Lambda$ which are distance $t$ apart then the size of $\Lambda$ is $O(t)$. As a consequence we improve a result of Iosevich and Jaming and show that $\Lambda$ has at most $O(R^{2/3})$ elements in any disk of radius $R$