Cesàro summability of Taylor series in weighted Dirichlet spaces

On Nörlund summability of Taylor series in weighted Dirichlet spaces

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 …

Power-series summability methods in de Branges-Rovnyak spaces

We show that there exists a de Branges-Rovnyak space $\mathcal{H}(b)$ on the unit disk containing a function $f$ with the following property: even though $f$ can be approximated by polynomials in $\mathcal{H}(b)$, neither the Taylor partial sums of $f$ nor their Cesàro, Abel, Borel or logarithmic means converge to $f$ in $\mathcal{H}(b)$. A key tool is a new abstract result showing that, if one regular summability method includes another for scalar sequences, then it automatically does so for certain Banach-space-valued sequences too.

核可和方法与 Silverman–Toeplitz 定理

Kernel-Summability Methods and the Silverman-Toeplitz Theorem

本文引入 Banach 空间中的核可和方法，建立其 Silverman–Toeplitz 型定理与标量到向量值情形的包含关系推广，并用于研究全纯函数 Taylor 级数的可和性。

关于de Branges-Rovnyak空间中Taylor级数的发散性

On the Divergence of Taylor Series in de Branges-Rovnyak Spaces

本文研究de Branges-Rovnyak空间中Taylor级数的发散性，证明部分和可在范数上趋于无穷，并将结果推广至Cesàro平均等求和法。

关于Borel可和性与解析泛函

On Borel summability and analytic functionals

证明了形式幂级数有正的收敛半径当且仅当它在圆心为原点的圆上是一致Borel可和的。因此，我们得到了整函数$f$是指数型的当且仅当形式幂级数$\sum_{n=0}^{\infty}f^{(n)}(0)z^{n}$在以原点为中心的圆上是一致Borel可和的.我们应用这些结果，以获得这些席尔瓦回火超分布，这是解析泛函的表征。我们还使用Borel求和表示解析泛函的Borel和他们的时刻泰勒级数在Borel多边形。

Riesz summability on boundary lines of holomorphic functions generated by Dirichlet series

A particular consequence of the famous Carleson-Hunt theorem is that the Taylor series expansions of bounded holomorphic functions on the open unit disk converge almost everywhere on the boundary, whereas on single points the convergence may fail. In contrast, Bayart, Konyagin, and Queffélec constructed an example of an ordinary Dirichlet series $\sum a_n n^{-s}$, which on the open right half-plane $[Re >0]$ converges pointwise to a bounded, holomorphic function -- but diverges at each point of the imaginary line, although its limit function extends continuously to the closed right half plane.…