Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim's sense
After a brief summary of Tauberian conditions for ordinary sequences of numbers, we consider summability of double sequences of real or complex numbers by weighted mean methods which are not necessarily products of related weighted mean methods in one variable. Our goal is to obtain Tauberian condit...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2004-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171204403329 |
| Summary: | After a brief summary of Tauberian conditions for
ordinary sequences of numbers, we consider summability of double
sequences of real or complex numbers by weighted mean methods
which are not necessarily products of related weighted mean
methods in one variable. Our goal is to obtain Tauberian
conditions under which convergence of a double sequence follows
from its summability, where convergence is understood in
Pringsheim's sense. In the case of double sequences of real numbers,
we present necessary and sufficient Tauberian conditions, which are so-called
one-sided conditions. Corollaries allow these Tauberian conditions
to be replaced by Schmidt-type slow decrease conditions.
For double sequences of complex numbers, we present necessary and
sufficient so-called two-sided Tauberian conditions.
In particular, these conditions are satisfied if the summable
double sequence is slowly oscillating. |
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| ISSN: | 0161-1712 1687-0425 |