In this paper, we solve a separation of singularities problem in the Bergman space. More precisely, we show that if $\mathcal{P}\subset \mathbb{C}$ is a convex polygon which is the intersection of n half planes, then the Bergman space on $\mathcal{P}$ decomposes into the sum of the Bergman spaces on these half planes. The result applies to the characterization of the reachable space of the one-dimensional heat equation on a finite interval with boundary controls. We prove that this space is a Bergman space of the square which has the given interval as a diagonal. This gives an affirmative answer to a conjecture raised in [17].

中文翻译：

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Bergman空间的奇异性分离及其在控制理论中的应用

在本文中，我们解决了Bergman空间中的奇点分离问题。更确切地说，我们表明$\mathcal{P}\subset \mathbb{C}$是一个凸多边形，它是n个半平面的交集，然后是$\mathcal{P}$分解为这些半平面上Bergman空间的总和。该结果适用于具有边界控制的有限区间上一维热方程可及空间的刻画。我们证明该空间是正方形的Bergman空间，其给定间隔为对角线。这为[17]中提出的猜想给出了肯定的答案。