Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

令 $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ 是定义在可数多个开放单位圆盘 ${\ mathbb {D}}\subset {{\mathbb C}}$ 。我们证明了 $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ 的稠密稳定秩是 $1$ 并且利用这个事实证明了一些非线性龙格型逼近定理 $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ 映射。然后我们应用这些结果来获得代数 $H^\infty ({\mathbb {D}})$ 的类似逼近问题中逼近映射范数的先验统一估计。