View source for Hölder summation methods

A collection of methods for summing series of numbers, introduced by O. Hölder [[#References|[1]]] as a generalization of the summation method of arithmetical averages (cf. [[Arithmetical averages, summation method of|Arithmetical averages, summation method of]]). The series

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is summable by the Hölder method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h0475182.png" /> to sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h0475183.png" /> if

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where

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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h0475187.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h0475188.png" />-summability of a series indicates that it converges in the ordinary sense; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h0475189.png" /> is the method of arithmetical averages. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751810.png" />-methods are totally [[Regular summation methods|regular summation methods]] for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751811.png" /> and are compatible for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751812.png" /> (cf. [[Compatibility of summation methods|Compatibility of summation methods]]). The power of the method increases with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751813.png" />: If a series is summable to a sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751814.png" /> by the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751815.png" />, it will also be summable to that sum by the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751816.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751817.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751818.png" /> the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751819.png" /> is equipotent and compatible with the Cesàro summation method of the same order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751820.png" /> (cf. [[Cesàro summation methods|Cesàro summation methods]]). If a series is summable by the method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751821.png" />, its terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751822.png" /> necessarily satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047518/h04751823.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Hölder,   "Grenzwerthe von Reihen an der Konvergenzgrenze"  ''Math. Ann.'' , '''20'''  (1882)  pp. 535–549</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Oxford Univ. Press  (1949)</TD></TR></table>