Lectures on multivariate polynomial approximation

Lectures on Multivariate Polynomial Interpolation

. Workshop on Function Approximation December 1 , 2016 – December 2 , 2016

s – Lectures Thursday, December 1, 9:00-10:30 Lecture 1: Spline approximation Stefan Takacs RICAM, Computational Methods for PDEs Thomas Takacs JKU Linz, Institute of Applied Geometry In the first part of the lecture we discuss standard approximation properties for B-spline and NURBS spaces. B-splines are piecewise polynomial functions of a given prescribed smoothness. NURBS stands for non-unif...

A method to obtain the best uniform polynomial approximation for the family of rational function

In this article, by using Chebyshev’s polynomials and Chebyshev’s expansion, we obtain the best uniform polynomial approximation out of P2n to a class of rational functions of the form (ax2+c)-1 on any non symmetric interval [d,e]. Using the obtained approximation, we provide the best uniform polynomial approximation to a class of rational functions of the form (ax2+bx+c)-1 for both cases b2-4a...

Lecture 04 / 13 : Graph Sampling and Sparsification

In previous lectures, we discussed various topics on graph: sparsest cut, clustering which finds small k-cuts, max flow, multicommodity flow, etc. For very large graphs, we’d like to ask “instead of regular approximation algorithms which run in polynomial time, can we find any near-linear time algorithm for minimum cut or maximum flow problems?”. A simple idea is for a given graph G, to solve t...

Factorization of multivariate positive Laurent polynomials

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. F...