发表机构
School of Mathematics, Hohai University（河海大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为具有正指数谱的多尺度核空间建立了端点稳定的区间逆不等式，通过保留对角能量和控制非对角项，得到与尺度数无关的系数界，并证明了尺度指数的锐性，为确定性采样稳定性提供了定量条件。
AI 中文摘要
逆不等式对于控制导数增长和确保核逼近中的稳定采样至关重要。现有的全空间和单尺度估计提供了已确立的分析基础，但它们在有限区间上的多尺度解析空间中的推广需要对边界效应和跨尺度抵消进行统一控制。本文针对由具有正指数谱的核的原函数生成的空间，建立了无权重区间\(L^2\)逆不等式。通过保留外部对角能量，并利用带端点距离权重的一侧估计来控制非对角相互作用，我们推导了归一化高阶导数的统一系数界。一个合成算子扰动论证将这些界推广到正谱混合，而相邻导数比较和区间插值产生阶为\(h_*^{-1}\)的逆界，其中\(h_*\)是最小尺度。在几何尺度分离和尺度内中心分离的条件下，该常数与尺度和中心的数量及其到端点的距离无关，允许端点中心和跨尺度重合中心。系数下界接近最优极限\(1/2\)。双中心构造确立了该类中每个固定核的尺度指数的锐性，并且对于单个指数谱，最佳均匀系数下界与\(1/2\)的偏差具有阶为\(q^{-1/2}\)的锐亏量，其中\(q\)是导数阶。针对柯西核和逻辑斯蒂核的数值示例展示了在多种多尺度配置下的系数稳定性。这些结果将区间逆估计推广到具有端点中心的多尺度解析核空间，并为确定性采样稳定性提供了定量条件。
英文摘要
Inverse inequalities are essential for controlling derivative growth and ensuring stable sampling in kernel approximation. Existing whole-space and single-scale estimates provide an established analytical basis, but their extension to multiscale analytic spaces on finite intervals requires uniform control of boundary effects and cross-scale cancellation. This paper establishes unweighted interval \(L^2\) inverse inequalities for spaces generated by primitives of kernels with positive exponential spectra. By retaining exterior diagonal energy and controlling off-diagonal interactions through one-sided estimates with endpoint-distance weights, we derive uniform coefficient bounds for normalized high derivatives. A synthesis-operator perturbation argument extends these bounds to positive spectral mixtures, while adjacent-derivative comparison and interval interpolation yield an inverse bound of order \(h_*^{-1}\), where \(h_*\) is the smallest scale. Under geometric scale separation and within-scale centre separation, the constant is independent of the number of scales and centres and their distances from the endpoints, allowing endpoint centres and coincident centres across scales. The coefficient lower bound approaches the optimal limit \(1/2\). Two-centre constructions establish the sharpness of the scale exponent for every fixed kernel in the class and, for a single exponential spectrum, a sharp deficit of order \(q^{-1/2}\) from \(1/2\) for the best uniform coefficient lower bound, where \(q\) is the derivative order. Numerical illustrations for Cauchy and logistic kernels demonstrate coefficient stability in several multiscale configurations. These results extend interval inverse estimates to multiscale analytic kernel spaces with endpoint centres and provide quantitative conditions for deterministic sampling stability.
