Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincaré inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincaré inequality, where the measurement function is of subsampled type, with a small but non-zero lengthscale that will be made precise. Our analysis identifies this inequality as a basic tool for function recovery problems. We discuss and demonstrate the optimality of the inequality concerning the subsampled lengthscale, connecting it to existing results in the literature. In application to function approximation problems, the approximation accuracy using different basis functions and under different regularity assumptions is established by using the subsampled Poincaré inequality. We observe that the error bound blows up as the subsampled lengthscale approaches zero, due to the fact that the underlying function is not regular enough to have well-defined pointwise values. A weighted version of the Poincaré inequality is proposed to address this problem; its optimality is also discussed.

中文翻译：

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通过二次采样Poincaré不等式进行函数逼近

长期以来，在广泛的应用数学和统计领域研究了通过一些采样数据进行函数逼近和恢复。诸如庞加莱不等式之类的分析工具可方便地估算不同尺度下的近似误差。本文的目的是研究广义庞加莱不等式，其中测量函数为二次采样类型，其长度尺度很小但不为零，将被精确化。我们的分析将这种不平等视为解决功能恢复问题的基本工具。我们讨论并证明了关于二次采样长度尺度的不等式的最优性，并将其与文献中的现有结果联系起来。在应用函数逼近问题时，通过使用二次采样的庞加莱不等式建立使用不同基函数和不同规律性假设的近似精度。我们观察到，由于底层函数的规则性不足以提供明确定义的逐点值，因此当二次采样的长度标度接近零时，误差范围会爆炸。提出了庞加莱不等式的加权形式来解决这个问题。还讨论了其最佳性。