View source for Kernel of a complex sequence

The set of points in the extended complex plane that for a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552401.png" /> is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552402.png" /> be the smallest closed convex domain in the complex plane that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552403.png" />. If there is no half-plane containing these points, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552404.png" /> is the whole complex plane, including the point at infinity; if such half-planes exist, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552405.png" /> is their common part. The point at infinity belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552406.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552407.png" /> is unbounded, and does not if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552408.png" /> is bounded. The intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k0552409.png" /> is called the kernel of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524010.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524011.png" /> is bounded, then its kernel coincides with the closed convex hull of the set of limit points; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524012.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524014.png" />, then the kernel is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524015.png" />. The kernel of a real sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524016.png" /> is the interval of the real line with end points:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524017.png" /></td> </tr></table>

The kernel of a sequence cannot be empty, although it may consist only of the point at infinity, as, for example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524018.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524019.png" />. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524020.png" /> with kernel consisting of the point at infinity is sometimes called definitely divergent. For a real sequence this means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524021.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524022.png" />.

Questions of kernel inclusion of summation methods are considered in the theory of summability. A summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524023.png" /> is kernel-stronger than a summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524024.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524025.png" /> of sequences if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524026.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524029.png" /> are, respectively, the kernels of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524031.png" />, that is, of sequences of averages of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055240/k05524032.png" />.

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