标题： Bergman空间中的高次简单部分分式：逼近与优化 显示英文标题

标题： High degree simple partial fractions in the Bergman space: Approximation and Optimization

摘要： 我们考虑标准加权Bergman空间的类$A^2_{\alpha}(\mathbb{D})$和具有单位圆上极点的简单分式集合$SF^N(\mathbb{T})$，其中这些分式的次数为$N$。我们证明，在某些条件下，单位圆上有$N$个极点且阶数为$n$的简单分式若具有最小范数，则这些点必在单位圆上等分布；反之亦然。我们指出，如果所施加的条件未被满足，这种情况将不再成立，并展示出一种新的有趣现象。我们还找到了这些范数的尖锐渐近表达式。此外，我们描述了这些分式在标准加权Bergman空间中的闭包。 显示英文摘要

摘要： We consider the class of standard weighted Bergman spaces $A^2_{\alpha}(\mathbb{D})$ and the set $SF^N(\mathbb{T})$ of simple partial fractions of degree $N$ with poles on the unit circle. We prove that under certain conditions, the simple partial fractions of order $N$, with $n$ poles on the unit circle attain minimal norm if and only if the points are equidistributed on the unit circle. We show that this is not the case if the conditions we impose are not met, exhibiting a new interesting phenomenon. We find sharp asymptotics for these norms. Additionally we describe the closure of these fractions in the standard weighted Bergman spaces.