发表机构
Uppsala University（乌普萨拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对原生空间为Sobolev空间的缩放正定核空间，建立了尺度一致逆不等式与Bernstein不等式，为核近似方法的多尺度稳定性分析提供了关键工具。
AI 中文摘要
逆不等式是核近似方法稳定性与收敛性分析的重要工具。然而在多尺度场景中，试验空间会随核尺度δ变化，固定核的逆估计并不充分，常数需在δ→0时仍保持可控。本文研究由缩放正定核生成的空间的尺度一致逆不等式，该空间的原生空间为Sobolev空间。我们首先建立依赖尺度的Sobolev空间中的逆估计，在有界域上，这会得到以L²为弱范数的标准Sobolev范数下的尺度一致逆不等式；该估计仅要求中心的分离距离不超过核尺度的固定倍数，这允许核尺度的减小速度慢于分离距离，与多尺度加密场景相关。随后我们在全空间上建立更通用的尺度一致Bernstein不等式，结果涵盖了在分离距离与核尺度相同关系下的大范围强弱Sobolev指数。证明直接使用核空间中函数的傅里叶表示，结合高低频分解与分离指数多项式的框架估计，避免了分别比较缩放与标准Sobolev范数产生的额外尺度因子。
英文摘要
Inverse inequalities are an important tool in the stability and convergence analysis of kernel approximation methods. In a multiscale setting, however, the trial space changes with the kernel scale $δ$, and inverse estimates for a fixed kernel are not sufficient. The constants must remain controlled as $δ\to0$. In this paper, we study scale-uniform inverse inequalities for spaces generated by scaled positive definite kernels whose native spaces are Sobolev spaces. We first establish inverse estimates in scale-dependent Sobolev spaces. On bounded domains, this leads to a scale-uniform inverse inequality in standard Sobolev norms with $L^2$ as the weaker norm. The estimate requires only that the separation distance of the centers be bounded above by a fixed multiple of the kernel scale. This allows the kernel scale to decrease more slowly than the separation distance, as is relevant in multiscale refinement. We then establish more general scale-uniform Bernstein inequalities on the whole space. Our result covers a broad range of weaker and stronger Sobolev indices under the same relation between the separation distance and the kernel scale. The proof works directly with the Fourier representation of functions in the kernel space and combines a low-high frequency decomposition with frame estimates for separated exponential polynomials. This avoids the additional scale factors that arise from separate comparisons of scaled and standard Sobolev norms.
