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On the exactness of a theorem of Mazur and Orlicz
N. A. Davydov Kiev State Pedagogical Institute
Abstract:
For a preassigned unbounded sequence $\{S_n\}$ of complex numbers, and preassigned complex numbers $z_1$ and $z_2\ne z_1$ we consruct: 1) regular matrices $A=||a_{nk}||$ and $B=||b_{nk}||$ such that the same bounded sequences are summable by these matrices and that $\lim\limits_{n\to\infty}S_n=z_1(A)$, $\lim\limits_{n\to\infty}S_n=z_2(B)$; 2) regular matrices $A^{(1)}=||a^{(1)}_{nk}||$ and $B^{(1)}=||b^{(1)}_{nk}||$ such that $B^{(1)}\subseteq A^{(1)}$, $\lim\limits_{n\to\infty}S_n=z_1(A^{(1)})$ and $\lim\limits_{n\to\infty}S_n=z_2(B^{(1)})$. Our results show that the well known theorem of Mazur–Orlicz on the bounded consistency of two regular matrices, one of which is boundedly stronger than the other, is exact.
Received: 18.02.1971
Citation:
N. A. Davydov, “On the exactness of a theorem of Mazur and Orlicz”, Mat. Zametki, 11:4 (1972), 431–436; Math. Notes, 11:4 (1972), 263–265
Linking options:
https://www.mathnet.ru/eng/mzm9807 https://www.mathnet.ru/eng/mzm/v11/i4/p431
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