This paper introduces a novel approach to approximating continuous functions over high-dimensional hypercubes by integrating matrix CUR decomposition with hyperinterpolation techniques. Traditional Fourier-based hyperinterpolation methods suffer from the curse of dimensionality, as the number of coefficients grows exponentially with the dimension. To address this challenge, we propose two efficient strategies for constructing low-rank matrix CUR decompositions of the coefficient matrix, significantly reducing computational complexity while preserving accuracy. The first method employs structured index selection to form a compressed representation of the tensor, while the second utilizes adaptive sampling to further optimize storage and computation. Theoretical error bounds are derived for both approaches, ensuring rigorous control over approximation quality. Additionally, practical algorithms -- including randomized and adaptive decomposition techniques -- are developed to efficiently compute the CUR decomposition. Numerical experiments demonstrate the effectiveness of our methods in drastically reducing the number of required coefficients without compromising precision. Our results bridge matrix/tensor decomposition and function approximation, offering a scalable solution for high-dimensional problems. This work advances the field of numerical analysis by providing a computationally efficient framework for hyperinterpolation, with potential applications in scientific computing, machine learning, and data-driven modeling.

翻译：本文提出了一种将矩阵CUR分解与超插值技术相结合的新方法，用于逼近高维超立方体上的连续函数。传统的基于傅里叶的超插值方法受制于维度灾难，因为系数数量随维度呈指数增长。为应对这一挑战，我们提出了两种高效策略来构建系数矩阵的低秩CUR分解，从而在保持精度的同时显著降低计算复杂度。第一种方法采用结构化指标选择来构建张量的压缩表示，第二种方法则利用自适应采样进一步优化存储和计算。我们为两种方法推导了理论误差界，确保对逼近质量进行严格控制。此外，本文还开发了包括随机化和自适应分解技术在内的实用算法，以高效计算CUR分解。数值实验表明，我们的方法能在不损失精度的情况下大幅减少所需系数的数量。本研究结果连接了矩阵/张量分解与函数逼近，为高维问题提供了可扩展的解决方案。这项工作通过为超插值提供一个计算高效的框架，推动了数值分析领域的发展，并在科学计算、机器学习和数据驱动建模中具有潜在应用前景。