Kernel of a complex sequence
The set of points in the extended complex plane that for a sequence is defined as follows. Let
be the smallest closed convex domain in the complex plane that contains
. If there is no half-plane containing these points, then
is the whole complex plane, including the point at infinity; if such half-planes exist, then
is their common part. The point at infinity belongs to
if
is unbounded, and does not if
is bounded. The intersection
is called the kernel of the sequence
.
If is bounded, then its kernel coincides with the closed convex hull of the set of limit points; if
converges to
,
, then the kernel is
. The kernel of a real sequence
is the interval of the real line with end points:
![]() |
The kernel of a sequence cannot be empty, although it may consist only of the point at infinity, as, for example, for where
. A sequence
with kernel consisting of the point at infinity is sometimes called definitely divergent. For a real sequence this means that
or
.
Questions of kernel inclusion of summation methods are considered in the theory of summability. A summation method is kernel-stronger than a summation method
on a set
of sequences if
for any
, where
and
are, respectively, the kernels of
and
, that is, of sequences of averages of
.
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) |
Kernel of a complex sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_complex_sequence&oldid=47488