论文标题
Tauberian定理用于普通融合
Tauberian theorems for ordinary convergence
论文作者
论文摘要
我们表明，实际序列$ x $是且仅当存在常规矩阵$ a $和$ f_ {σδ} $ - 理想$ \ natercal {i} $上的$ \ m arthbf {n} $上的$ \ mathbf {n} $，以便将$ y yy $ yy $ yy as $ y y yathcal as carty carty of ass of yath $ \ y y as carty ass的$ \ mathbf {n} $。这包括$ \ Mathcal {i} $是渐近密度零集的理想，Banach密度零集的理想和有限集的理想的情况。后者恢复了Keogh和Petersen在[J.伦敦数学。 Soc。 \ textbf {33}（1958），121--123]。我们的证明具有不同的性质，并依赖于$ \ Mathcal {i} $ - Baire类和过滤游戏的最新结果。 作为应用程序，我们获得了经典的Steinhaus定理的更强版本：对于每个常规矩阵$ a $，都存在$ \ {0,1 \} $ - 有价值的序列$ x $，因此$ ax $不是统计上的收敛性。
We show that a real sequence $x$ is convergent if and only if there exist a regular matrix $A$ and an $F_{σδ}$-ideal $\mathcal{I}$ on $\mathbf{N}$ such that the set of subsequences $y$ of $x$ for which $Ay$ is $\mathcal{I}$-convergent is of the second Baire category. This includes the cases where $\mathcal{I}$ is the ideal of asymptotic density zero sets, the ideal of Banach density zero sets, and the ideal of finite sets. The latter recovers an old result given by Keogh and Petersen in [J. London Math. Soc. \textbf{33} (1958), 121--123]. Our proofs are of a different nature and rely on recent results in the context of $\mathcal{I}$-Baire classes and filter games. As application, we obtain a stronger version of the classical Steinhaus' theorem: for each regular matrix $A$, there exists a $\{0,1\}$-valued sequence $x$ such that $Ax$ is not statistically convergent.
