2016/07/28 by Cordian Riener, Riener, Cordian, Markus Schweighofer +1 · 2 citations
Mathematics · #14H50 #14P05 (Secondary) #65D32 (Primary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical Approximation and Integration #Mathematical functions and polynomials #Numerical Analysis (math.NA) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1607.08404
openalex publication_date 2016/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let d and k be positive integers. Let \μ be a positive Borel measure\non \ℝ2 possessing finite moments up to degree 2d-1. If the support\nof \μ is contained in an algebraic curve of degree k, then we show that\nthere exists a quadrature rule for \μ with at most dk many nodes all\nplaced on the curve (and positive weights) that is exact on all polynomials of\ndegree at most 2d-1. This generalizes both Gauss and (the odd degree case of)\nSzeg Ho quadrature where the curve is a line and a circle, respectively, to\narbitrary plane algebraic curves. We use this result to show that, without any\nhypothesis on the support of \μ, there is always a cubature rule for \μ\nwith at most frac32d(d-1)+1 many nodes. In both results, we show that the\nquadrature or cubature rule can be chosen such that its value on a certain\npositive definite form of degree 2d is minimized. We characterize the unique\nGaussian quadrature rule on the line as the one that minimizes this value or\nseveral other values as for example the sum of the nodes' distances to the\norigin. The tools we develop should prove useful for obtaining similar results\nin higher-dimensional cases although at the present stage we can present only\npartial results in that direction.\n