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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
 
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
  
<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/h/h047/h047518/h0475181.png" /></td> </tr></table>
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$$\sum_{n=0}^\infty a_n$$
  
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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is summable by the Hölder method $(H,k)$ to sum $s$ 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/h/h047/h047518/h0475184.png" /></td> </tr></table>
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$$\lim_{n\to\infty}H_n^k=s,$$
  
 
where
 
where
  
<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/h/h047/h047518/h0475185.png" /></td> </tr></table>
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$$H_n^0=s_n=\sum_{k=0}^na_k,$$
  
<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/h/h047/h047518/h0475186.png" /></td> </tr></table>
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$$H_n^k=\frac{H_0^{k-1}+\ldots+H_n^{k-1}}{n+1},$$
  
<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" />.
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$k=1,2,\ldots$. In particular, $(H,0)$-summability of a series indicates that it converges in the ordinary sense; $(H,1)$ is the method of arithmetical averages. The $(H,k)$-methods are totally [[Regular summation methods|regular summation methods]] for any $k$ and are compatible for all $k$ (cf. [[Compatibility of summation methods|Compatibility of summation methods]]). The power of the method increases with increasing $k$: If a series is summable to a sum $s$ by the method $(H,k)$, it will also be summable to that sum by the method $(H,k')$ for any $k'>k$. For any $k$ the method $(H,k)$ is equipotent and compatible with the Cesàro summation method of the same order $k$ (cf. [[Cesàro summation methods|Cesàro summation methods]]). If a series is summable by the method $(H,k)$, its terms $a_n$ necessarily satisfy the condition $a_n=o(n^k)$.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 17:32, 30 July 2014

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

$$\sum_{n=0}^\infty a_n$$

is summable by the Hölder method $(H,k)$ to sum $s$ if

$$\lim_{n\to\infty}H_n^k=s,$$

where

$$H_n^0=s_n=\sum_{k=0}^na_k,$$

$$H_n^k=\frac{H_0^{k-1}+\ldots+H_n^{k-1}}{n+1},$$

$k=1,2,\ldots$. In particular, $(H,0)$-summability of a series indicates that it converges in the ordinary sense; $(H,1)$ is the method of arithmetical averages. The $(H,k)$-methods are totally regular summation methods for any $k$ and are compatible for all $k$ (cf. Compatibility of summation methods). The power of the method increases with increasing $k$: If a series is summable to a sum $s$ by the method $(H,k)$, it will also be summable to that sum by the method $(H,k')$ for any $k'>k$. For any $k$ the method $(H,k)$ is equipotent and compatible with the Cesàro summation method of the same order $k$ (cf. Cesàro summation methods). If a series is summable by the method $(H,k)$, its terms $a_n$ necessarily satisfy the condition $a_n=o(n^k)$.

References

[1] O. Hölder, "Grenzwerthe von Reihen an der Konvergenzgrenze" Math. Ann. , 20 (1882) pp. 535–549
[2] G.H. Hardy, "Divergent series" , Oxford Univ. Press (1949)
How to Cite This Entry:
Hölder summation methods. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=H%C3%B6lder_summation_methods&oldid=18982
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article