Kernels of sequences of complex numbers and their regular transformations

A. A. Shcherbakov

Matematičeskie zametki, Tome 22 (1977) no. 6, pp. 815-823.

Résumé

It is proved that $\bigcap\limits_xU(x,C\varlimsup\limits_{n\to\infty}|x-x_n|)$, where $U(a,r)$ is the ball of radius $r$ with center at the pointa, is the smallest closed convex set containing the kernel of any sequence $\{y_n\}$ obtained from the sequence $\{x_n\}$ by means of a regular transformation $(c_{nk})$, satisfying the condition $\varlimsup\limits_{n\to\infty}\sum_{k=1}^\infty|c_{kn}|=C\ge1$, where $x$, $x_n$, $c_{nk}$, ($n,k=1,2,\dots$) are complex numbers.