加权Dirichlet空间中Taylor级数的Nörlund可和性
On Nörlund summability of Taylor series in weighted Dirichlet spaces
摘要：
在本文中，我们证明了加权狄利克雷空间中函数的泰勒级数是（广义）Nörlund可和的，前提是确定Nörlund算子的序列是非减的并且具有有限的上增长率。特别是泰勒级数是Nörlund可求和的所有$α>1/2$，收敛速率的顺序是$O(n^{-1/2})$。不等式$α>1/2$是尖锐的。另一方面，如果泰勒级数是Nörlund可和的，并且决定序列的部分和满足一定的增长条件，则决定序列具有有限的下增长率。一个类似的结果是来自一个非增序列，一致有界远离零。
英文摘要：
In this note we show that the Taylor series of a function in a weighted Dirichlet space is (generalized) Nörlund summable, provided that the sequence determining the Nörlund operator is non-decreasing and has finite upper growth rate. In particular the Taylor series is Nörlund summable for all $α>1/2$, and the rate of convergence is of the order $O(n^{-1/2})$. The inequality $α>1/2$ is sharp. On the other hand if the Taylor series is Nörlund summable and the partial sums of the determining sequence enjoy a certain growth condition then the determining sequence has finite lower growth rate. An analogue result is derived for a non-increasing sequence that is uniformly bounded away from zero.
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