In the current note we consider matrix-sequences $\{B_{n,t}\}_n$ of increasing sizes depending on $n$ and equipped with a parameter $t>0$. For every fixed $t>0$, we assume that each $\{B_{n,t}\}_n$ possesses a canonical spectral/singular values symbol $f_t$ defined on $D_t\subset \R^{d}$ of finite measure, $d\ge 1$. Furthermore, we assume that $ \{ \{ B_{n,t}\} : \, t > 0 \} $ is an approximating class of sequences (a.c.s.) for $ \{ A_n \} $ and that $ \bigcup_{t > 0} D_t = D $ with $ D_{t + 1} \supset D_t $. Under such assumptions and via the notion of a.c.s, we prove results on the canonical distributions of $ \{ A_n \} $, whose symbol, when it exists, can be defined on the, possibly unbounded, domain $D$ of finite or even infinite measure. We then extend the concept of a.c.s. to the case where the approximating sequence $ \{ B_{n,t}\}_n $ has possibly a different dimension than the one of $ \{ A_n\} $. This concept seems to be particularly natural when dealing, e.g., with the approximation both of a partial differential equation (PDE) and of its (possibly unbounded, or moving) domain $D$, using an exhausting sequence of domains $\{ D_t \}$. Examples coming from approximated PDEs/FDEs with either moving or unbounded domains are presented in connection with the classical and the new notion of a.c.s., while numerical tests and a list of open questions conclude the present work.

翻译：在本研究中，我们考虑矩阵序列 $\{B_{n,t}\}_n$，其维度随 $n$ 递增且依赖于参数 $t>0$。对于任意固定的 $t>0$，我们假设每个 $\{B_{n,t}\}_n$ 在有限测度区域 $D_t\subset \R^{d}$（$d\ge 1$）上具有规范谱/奇异值符号 $f_t$。进一步假设 $\{ \{ B_{n,t}\} : \, t > 0 \}$ 构成 $\{ A_n \}$ 的渐近循环序列近似类，且满足 $\bigcup_{t > 0} D_t = D$ 与 $D_{t + 1} \supset D_t$。基于渐近循环序列的概念，我们在上述假设条件下证明了关于 $\{ A_n \}$ 规范分布的结果——当符号存在时，其可定义在可能无界但具有有限（甚至无限）测度的区域 $D$ 上。随后，我们将渐近循环序列的概念推广至近似序列 $\{ B_{n,t}\}_n$ 与目标序列 $\{ A_n\}$ 维度可能不同的情形。这一推广在处理偏微分方程及其（可能无界或移动的）定义域 $D$ 的近似问题时显得尤为自然，例如通过使用区域穷竭序列 $\{ D_t \}$ 进行逼近。本文结合经典与新提出的渐近循环序列概念，给出了涉及移动或无界区域的偏微分方程/分数阶微分方程近似问题的具体算例，并通过数值实验与开放性问题列表对研究工作进行了总结。