Band-limited functions are fundamental objects that are widely used in systems theory and signal processing. In this paper we refine a recent nonparametric, nonasymptotic method for constructing simultaneous confidence regions for band-limited functions from noisy input-output measurements, by working in a Paley-Wiener reproducing kernel Hilbert space. Kernel norm bounds are tightened using a uniformly-randomized Hoeffding's inequality for small samples and an empirical Bernstein bound for larger ones. We derive an approximate threshold, based on the sample size and how informative the inputs are, that governs which bound to deploy. Finally, we apply majority voting to aggregate confidence sets from random subsamples, boosting both stability and region size. We prove that even per-input aggregated intervals retain their simultaneous coverage guarantee. These refinements are also validated through numerical experiments.

翻译：带限函数是系统理论与信号处理中广泛使用的基本对象。本文通过在一个Paley-Wiener再生核希尔伯特空间中工作，改进了一种从含噪输入输出测量构建带限函数同时置信区域的非参数、非渐近方法。我们利用适用于小样本的均匀随机化霍夫丁不等式与适用于大样本的经验伯恩斯坦界，收紧核范数界。基于样本量及输入信息量，我们推导出决定采用何种界限的近似阈值。最后，我们应用多数投票方法聚合来自随机子样本的置信集，从而提升稳定性并扩大区域范围。我们证明即使针对每个输入的聚合区间仍保持其同时覆盖保证。这些改进也通过数值实验得到验证。