AI 中文总结
该研究针对带噪数据下Sobolev空间函数的稳定谱微分问题，引入容许傅里叶乘子，证明正则化微分算子达极小极大最优稳定性速率，扩展了相关结果并验证了多种乘子的表现。
AI 中文摘要
我们研究从带噪数据中对Sobolev空间内函数进行稳定谱微分的问题。我们引入一类满足简单且可直接验证条件的容许傅里叶乘子，证明对应的正则化微分算子可达到极小极大最优稳定性速率。该结果将此前基于L²的结论扩展至Sobolev空间H^{s,p}(ℝⁿ)（1<p<∞）。分析依赖乘子估计，适用于高斯、谱截断、Tikhonov型正则化等多种乘子，数值示例展示了多种容许谱乘子的表现。
英文摘要
We study the problem of stable spectral differentiation of functions in Sobolev spaces from noisy data. We introduce a class of admissible Fourier multipliers under simple and directly verifiable conditions and show that the corresponding regularized differentiation operators achieve minimax optimal stability rates. The results extend the previous $L^2$ based results to Sobolev spaces $H^{s,p}(\mathbb{R}^n)$, $1<p<\infty$. The analysis relies on multiplier estimates and applies to a wide class of multipliers, including Gaussian, spectral cutoff, and Tikhonov-type regularizations. Numerical examples demonstrate the behavior of several admissible spectral multipliers.
Comments21 pages, 3 figures
