发表机构
Department of Mathematics, Faculty of Science, Istanbul University; Istinye University; Institute of Science, Istanbul University（伊斯坦布尔大学理学院数学系; 伊斯蒂耶大学; 伊斯坦布尔大学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 Stieltjes 时钟下超 g-Hölder 正则函数的原子结构，给出精确极值与时钟恢复方法，并应用于微分方程以量化跳跃行为。
AI 中文摘要
与普通时间不同，Stieltjes 时钟可能连续前进、在区间上保持恒定或发生跳跃。对于普通的连续时钟，指数 $\alpha>1$ 的 Hölder 正则性强制其恒定性，而 Stieltjes 时钟的跳跃则可能允许非恒定行为。我们将相对于 g 测量的指数 $\alpha>1$ 的 Hölder 正则性称为超 g-Hölder 正则性，并证明在有限维空间中，此类函数具有完全由其跳跃决定的原子表示，其 g-Hölder 半范数恰好由相应的归一化跳跃大小给出。这种原子表示给出了相关函数空间的线性等距描述，连同其 Banach 性质和可分性，一个具有最优界的精确全变差公式，以及有限跳跃逼近结果。然后我们将此结构应用于 Stieltjes 微分方程。非零状态跳跃需要正的时钟跳跃，导致定量的有限跳跃界，并且对于仿射动力学，在给定的总 Stieltjes 质量下，跳跃次数和极值终端距离具有精确阈值。最后，我们研究从观测到的状态跳跃中恢复 Stieltjes 时钟。对于已知的单值动力学，在自然的局部可识别性条件下可以恢复时钟跳跃，而对于非凸微分包含，超 g-Hölder 界可以减少、在某些情况下消除非唯一性。
英文摘要
Unlike ordinary time, a Stieltjes clock may advance continuously, remain constant over intervals, or jump. For an ordinary continuous clock, Hölder regularity with exponent $α>1$ forces constancy, whereas jumps of a Stieltjes clock may allow nonconstant behavior. We refer to Hölder regularity of exponent $α>1$ measured relative to g as super-g-Hölder regularity, and show that, in finite dimensions, such functions admit an atomic representation determined entirely by their jumps, with their g-Hölder seminorms given exactly by the corresponding normalized jump sizes. This atomic representation yields a linear isometric description of the associated function space, together with its Banach and separability properties, an exact total-variation formula with an optimal bound, and finite-jump approximation results. We then apply this structure to Stieltjes differential equations. Nonzero state jumps require positive clock jumps, leading to quantitative finite-jump bounds and, for affine dynamics, exact thresholds for the number of jumps and extremal terminal distances under a prescribed total Stieltjes mass. Finally, we study recovery of the Stieltjes clock from observed state jumps. For known single-valued dynamics, clock jumps can be recovered under a natural local identifiability condition, while for nonconvex differential inclusions the super-g-Hölder bound can reduce, and in some cases remove, nonuniqueness.
Comments29 pages
