AI 中文总结
研究希尔伯特空间中邻近算子相关系统的精确解存在唯一性，建立近似解充要条件，引入邻近逆性质并推导其刻画，将结果应用于可行性与信号恢复问题，拓展了优化与逆问题的研究框架。
AI 中文摘要
设$\boldsymbol{\textit{H}}$为实希尔伯特空间，$(f_i)_{i \text{∈} I}$是$\boldsymbol{\textit{H}}$上的真、下半连续且凸函数的有限族。本研究探讨形如$(\forall i \text{∈} I)\boldsymbol{\textit{prox}}_{f_i}(x)=p_i$的邻近算子相关系统的精确解的存在性与唯一性，其中$(p_i)_{i \text{∈} I}$是$\boldsymbol{\textit{H}}$中给定的邻近点集合，该系统自然推广了经典投影问题。我们建立了此类系统近似解存在的充要条件，还引入并推导了邻近逆性质（IPP）的若干刻画，其为最佳逼近逆性质（IBAP）的推广。所得结果在可行性问题与信号恢复问题中给出应用，证明了所提框架对希尔伯特空间中优化与逆问题的适用性。
英文摘要
Let $\HH$ be a real Hilbert space and let $(f_i)_{i\in I}$ be a finite family of proper, lower semicontinuous, and convex functions on $\HH$. This study investigates the existence and uniqueness of exact solutions to systems involving proximity operators of the form: \ $(\forall i\in I)\ \pr{f_i}(x)=p_i,$ where $(p_i)_{i\in I}$ is a prescribed collection of proximal points in $\HH$, which naturally generalize classical projection problems. We establish necessary and sufficient conditions for the existence of approximate solutions to such systems. Moreover, we introduce and derive several characterizations of the inverse proximal property (IPP), as a generalization of the inverse best approximation property (IBAP). Applications of the obtained results are presented in the contexts of a feasibility problem and signal recovery problem, demonstrating the relevance of the proposed framework to optimization and inverse problems in Hilbert spaces.
