卷积不变线性泛函及其在可和性方法中的应用
Convolution invariant linear functionals and applications to summability methods
摘要：
我们研究了$L^{\infty}(\mathbb{R})$上的拓扑不变平均，证明了关于一个卷积算子的不变性足以使一个平均成为拓扑不变的。我们也考虑这个结果的可和性方法的一些应用。特别地，对$L^{\infty}(\mathbb{R})$中的函数引入了几乎收敛的概念，得到了关于几乎收敛的Tauberian定理和由Wiener核定义的可和性方法.进一步，对于由Hölder可和性方法的极限定义的$C_{\infty}$可和性方法，我们给出了给定函数是$C_{\infty}$可和的充要条件.
英文摘要：
We study topologically invariant means on $L^{\infty}(\mathbb{R})$, the set of all essentially bounded functions on the real line, and prove that invariance with respect to a single convolution operator is sufficient for a mean to be topologically invariant. We also consider some applications of this result to summability methods. In particular, the notion of almost convergence is introduced for a function in $L^{\infty}(\mathbb{R})$, and a Tauberian theorem concerning almost convergence and a summability method defined by a Wiener kernel is obtained. Further, for the $C_{\infty}$ summability method, which is defined by the limit of Hölder summability methods, we provide a necessary and sufficient condition for a given function to be $C_{\infty}$ summable.
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