Сходимость подпосле довательности и сумм ируемость последовательности

Abstract

LetT be a regular positive method of summation,

$$n_k \in Z_ + , 0 = n_0< n_1< ...< n_k< ..., \mu _n \geqq 0 \left( {n \in N} \right).$$

Then an arbitrary sequence {S _n ,S _n ∈C (n∈Z ₊), for which

$$S_{n_k } \to S \in C \left( {k \to \infty } \right), \left| {S_n - S_{n - 1} } \right| \leqq \mu _n \left( {n \in N} \right),$$

is summable by methodT toS if and only if the sequence

$$t_n = \min \left\{ {\mathop \Sigma \limits_{v = n_k + 1}^n \mu _v , \mathop \Sigma \limits_{v = n + 1}^{n_{k + 1} } \mu _v } \right\} \left( {n \in Z_ + } \right)$$

is summable byT to 0.

This result gives a complete solution of a problem recently raised by Menšov [1].