In this paper, we introduce a method known aspolynomial frame approximationfor approximating smooth, multivariate functions defined on irregular domains in$d$dimensions, where$d$can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the correspondingNikolskii inequalityfor the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.

中文翻译：

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不规则域上的近似平滑、多变量函数

在本文中，我们介绍了一种称为多项式框架逼近用于逼近定义在不规则域上的平滑、多元函数$d$尺寸，其中$d$可以是任意的。这种方法很简单，并且仅依赖于边界张量积域上的正交多项式。特别是，不需要事先知道函数的域。当限制为子域时，正交基不再是基，而是框架。由于截断近似系统的近线性相关性，使用框架进行数值计算存在潜在困难。然而，可以通过正则化获得条件良好的近似值，例如截断奇异值分解。我们在本文中全面分析了这种近似，为具有经典和混合 Sobolev 正则的函数提供了误差估计，后者特别适用于高维问题。Nikolskii 不等式为域。特别是，我们表明，从统一度量中提取的点的样本复杂度在多项式空间的维度上是二次的（高达一个对数因子），独立于$d$，对于一大类非平凡的域。这扩展了超立方体中多项式逼近的一个众所周知的结果。