Semi-algebraic Approximation Using Christoffel–Darboux Kernel

• Published in: Constructive approximation Vol. 54; no. 3; pp. 391 - 429
• Main Authors: Marx, Swann, Pauwels, Edouard, Weisser, Tillmann, Henrion, Didier, Lasserre, Jean Bernard
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V; Springer Verlag
• Subjects: Algebra; Analysis; Approximation; Convergence; Evaluation; Gibbs phenomenon; Kernels; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical integration; Optimal control; Optimization and Control; Polynomials; Positive polynomials; Convex optimization; 41A30; Orthogonal polynomials; 47B32; Approximation theory; Moments; 42C05; 41A30 Contents; approximation theory; moments; convex optimization; positive polynomials; orthogonal polynomials 42C05; orthogonal polynomials
• ISSN: 0176-4276; 1432-0940
• DOI: 10.1007/s00365-021-09535-4

We provide a new method to approximate a (possibly discontinuous) function using Christoffel–Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy solutions to nonlinear hyperbolic PDEs, or using numerical integration from finitely many evaluations of the function. While most of the existing methods construct a piecewise polynomial approximation, we construct a semi-algebraic approximation whose estimation and evaluation can be performed efficiently. An appealing feature of this method is that it deals with nonsmoothness implicitly so that a single scheme can be used to treat smooth or nonsmooth functions without any prior knowledge. On the theoretical side, we prove pointwise convergence almost everywhere as well as convergence in the Lebesgue one norm under broad assumptions. Using more restrictive assumptions, we obtain explicit convergence rates. We illustrate our approach on various examples from control and approximation. In particular, we observe empirically that our method does not suffer from the Gibbs phenomenon when approximating discontinuous functions.