关于凸序列的刻画、分解与稳定性
On characterizations, Decompositions, and Stability of Convex Sequences
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中文总结 AI 辅助
本文研究凸序列的刻画、分解与稳定性，证明凸序列的等价条件，给出序列分解为两凸序列之差的结果，建立近似凸序列的Hyers-Ulam型稳定性定理，并探讨凸性与次可加性等的关联。
中文摘要 AI 辅助
本文针对凸序列引入了新的刻画、分解定理及稳定性结果。我们证明：一个序列是凸序列当且仅当其上方图满足中点凸性条件，从而将凸性的离散概念与几何概念关联起来。一项分解结果表明，任意序列均可表示为两个凸序列之差，并可推广到高阶凸性的情形。我们为下方有界序列构造了非平凡凸下确界，并建立了Hyers-Ulam型稳定性定理，该定理表明：任意近似凸序列均可被一个真正的凸序列一致逼近，且不会显著改变其取值。最后，针对凹序列，我们通过提供斜率不等式与单调辅助序列，刻画了其中为凸序列的子序列，探讨了凸性、次可加性及周期索引子序列之间的相互作用。
英文摘要
This paper introduces new characterizations, decomposition theorems, and stability results for convex sequences. We show that a sequence is convex precisely when its epigraph satisfies a midpoint convexity condition, thereby connecting discrete and geometric notions of convexity. A decomposition result proves that any sequence can be written as the difference of two convex sequences, with generalizations to higher-order convexity. We construct nontrivial convex minorants for bounded-below sequences and establish a Hyers-Ulam-type stability theorem showing that any approximately convex sequence can be uniformly approximated by a genuine convex sequence without significantly altering its values. Finally, for a concave sequence, we characterize those subsequences that are convex in it by providing slope inequalities and monotone auxiliary sequences. We explore the interplay among convexity, subadditivity, and periodically indexed subsequences.
