Numerical hyperinterpolation over nonstandard planar regions

We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes

Bivariate Lagrange interpolation at the node points of Lissajous curves - the degenerate case

In this article, we study bivariate polynomial interpolation on the node points of degenerate Lissajous figures. These node points form Chebyshev lattices of rank $1$ and are generalizations of the well-known Padua points. We show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the Lagrange basis polynomials. Further, we prove mean and uniform convergence of the interpolating schemes. For the uniform convergence the growth of the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature.Comment: 26 pages, 6 figures, 1 tabl

One-sided approximation in L of the characteristic function of an interval by trigonometric polynomials

The value of the best one-sided integral approximation of the characteristic function of the interval (-h, h) by trigonometric polynomials of given degree is found for any 0 < h ≤ π. © 2013 Pleiades Publishing, Ltd