Kernels of sequences of complex numbers and their regular transformations

Abstract

It is proved that$\mathop \cap \limits_x U(x,c\mathop {\overline {\lim } }\limits_{n \to \infty } |x - x_n |$, 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$\mathop {\overline {\lim } }\limits_{n \to \infty } \sum\nolimits_{k = 1}^\infty {|c_{nk} |} = c \geqslant 1$, where x, x_n, c_nk (n, k=1, 2,...) are complex numbers.