# Semi-algebraic approximation using Christoffel-Darboux kernel

## TL;DR

This paper introduces a semi-algebraic approximation method using Christoffel-Darboux kernels that effectively handles discontinuous functions, with proven convergence properties and practical advantages over traditional piecewise polynomial methods.

## Contribution

The paper presents a novel semi-algebraic approximation technique based on Christoffel-Darboux kernels, capable of approximating non-smooth functions without prior smoothness assumptions.

## Key findings

- Converges pointwise almost everywhere
- Achieves convergence in Lebesgue norm
- Does not suffer from Gibbs phenomenon in discontinuous cases

## Abstract

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 non-linear 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 non-smoothness implicitly so that a single scheme can be used to treat smooth or non-smooth 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 the Gibbs phenomenon when approximating discontinuous functions.