View source for Compatibility of summation methods

A property of summation methods that describes the consistency of the results of applying these methods. Two methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236602.png" /> are compatible if they cannot sum the same sequence or series to different limits, otherwise they are called incompatible summation methods. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236604.png" /> be summation methods of sequences say, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236606.png" /> be their summability fields. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c0236608.png" /> are compatible if

<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/c/c023/c023660/c0236609.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366012.png" /> are the numbers to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366013.png" /> is summed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366015.png" />, respectively. For example, all the [[Cesàro summation methods|Cesàro summation methods]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366016.png" /> are compatible for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366017.png" />, and so is every regular [[Voronoi summation method|Voronoi summation method]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366018.png" /> is some set of sequences and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366019.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366020.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366022.png" /> are compatible on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366023.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366025.png" /> are said to be completely compatible (for real sequences) if (*) also holds in case one includes in their summability fields the sequences summable by these methods to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023660/c02366027.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR></table>



====Comments====
The Voronoi summation methods are in the Western literature called Nörlund summation methods, despite the fact that G.F. Voronoi was the first to introduce these methods (1901). N.E. Nörlund later (1920), independently, rediscovered them.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Zeller,   W. Beekmann,   "Theorie der Limitierungsverfahren" , Springer  (1970)</TD></TR></table>