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.

我们提供了一种使用Christoffel–Darboux内核近似（可能不连续）函数的新方法。我们对未知多元函数的了解是基于函数图上支持的Young度量的有限矩。当逼近最佳控制问题的弱（或度量值）解决方案，非线性双曲型PDE的熵解或使用函数的有限多次评估的数值积分时，此类输入可用。尽管大多数现有方法都构造了分段多项式逼近，但我们构造了半代数逼近，其估计和评估可以有效地执行。该方法的一个吸引人的特征是它隐式处理了非平滑性，因此无需任何先验知识即可使用单个方案来处理平滑或非平滑函数。从理论上讲，我们证明了在广泛的假设下几乎所有地方都具有逐点收敛性以及Lebesgue一规范中的收敛性。使用更严格的假设，我们可以获得明确的收敛速度。我们从控制和逼近的各种示例中说明了我们的方法。特别地，我们凭经验观察到，当逼近不连续函数时，我们的方法不会遭受吉布斯现象的困扰。我们获得了明确的收敛速度。我们从控制和逼近的各种示例中说明了我们的方法。特别地，我们凭经验观察到，当逼近不连续函数时，我们的方法不会遭受吉布斯现象的困扰。我们获得了明确的收敛速度。我们从控制和逼近的各种示例中说明了我们的方法。特别地，我们凭经验观察到，当逼近不连续函数时，我们的方法不会遭受吉布斯现象的困扰。