Sequence transformations that guarantee a given rate of convergence (Q1119845)

Let t be a positive number sequence and define the sequence space \(\Omega (t):=\{x:\) \(x_ k=O(t_ k)\}\). Characterizations are given for summability matrices that map the spaces \(\ell^ 1\), \(\ell^{\infty}\), c, or \(c_ 0\) into \(\Omega\) (t), thus ensuring that the transformed sequences converge at least as fast as t. These characterizations are explicit conditions on the rows of the matrix. For example, the matrix A maps \(\ell^ 1\) into \(\Omega\) (t) if and only if \(\mu\) is in \(\Omega\) (t), where \(\mu_ n:=\sum^{\infty}_{k=1}| a_{nk}|\).