Kernel of a complex sequence

The set of points in the extended complex plane that for a sequence $ \{ z _ {n} \} $ is defined as follows. Let $ R _ {n} $ be the smallest closed convex domain in the complex plane that contains $ z _ {n + 1 } , z _ {n + 2 } , . . . $. If there is no half-plane containing these points, then $ R _ {n} $ is the whole complex plane, including the point at infinity; if such half-planes exist, then $ R _ {n} $ is their common part. The point at infinity belongs to $ R _ {n} $ if $ \{ z _ {n} \} $ is unbounded, and does not if $ \{ z _ {n} \} $ is bounded. The intersection $ K = \cap _ {n = 1 } ^ \infty R _ {n} $ is called the kernel of the sequence $ \{ z _ {n} \} $.

If $ \{ z _ {n} \} $ is bounded, then its kernel coincides with the closed convex hull of the set of limit points; if $ \{ z _ {n} \} $ converges to $ z _ {0} $, $ z _ {0} \neq \infty $, then the kernel is $ z _ {0} $. The kernel of a real sequence $ \{ z _ {n} \} $ is the interval of the real line with end points:

$$ a = \overline{\lim\limits}\; _ {n \rightarrow \infty } z _ {n} ,\ \ b = \overline{\lim\limits}\; _ {n \rightarrow \infty } z _ {n} . $$

The kernel of a sequence cannot be empty, although it may consist only of the point at infinity, as, for example, for $ \{ z _ {n} \} $ where $ z _ {n} = n + in $. A sequence $ \{ z _ {n} \} $ with kernel consisting of the point at infinity is sometimes called definitely divergent. For a real sequence this means that $ z _ {n} \rightarrow + \infty $ or $ z _ {n} \rightarrow - \infty $.

Questions of kernel inclusion of summation methods are considered in the theory of summability. A summation method $ A $ is kernel-stronger than a summation method $ B $ on a set $ U $ of sequences if $ K _ {A} \subset K _ {B} $ for any $ \{ z _ {n} \} \subset U $, where $ K _ {A} $ and $ K _ {B} $ are, respectively, the kernels of $ A $ and $ B $, that is, of sequences of averages of $ \{ z _ {n} \} $.

References

[1a]	K. Knopp, "Zur Theorie des Limitierungsverfahren I" Math. Z. , 31 (1930) pp. 97–127
[1b]	K. Knopp, "Zur Theorie des Limitierungsverfahren II" Math. Z. , 31 (1930) pp. 276–305
[2]	R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[3]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)