Semi-algebraic Approximation Using Christoffel–Darboux Kernel

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