The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rules at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and capable of mitigating the curse of dimensionality for approximating high-dimensional functions.

翻译：本文提出了一种用于高维函数逼近的通用拟插值方案。为便于误差分析，我们将拟插值视为两步过程：第一步通过特制的卷积算子逼近目标函数（其误差项称为卷积误差）；第二步在给定采样点处使用特定求积法则离散化该卷积算子（其误差项称为离散化误差）。最终逼近误差可表示为这两个误差项的最优平衡和，这亦将我们的拟插值视为平衡卷积误差与离散化误差的正则化技术。作为具体实例，我们构建了用于高维函数逼近的稀疏网格拟插值方案。理论分析与数值实验均证明，该拟插值方案具有鲁棒性，并能有效缓解高维函数逼近中的维度灾难问题。