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''permanent summation methods''
 
''permanent summation methods''
  
Methods for summing series (sequences) that sum every convergent series (sequence) to the same sum as that to which it converges. Regular summation methods are a special case of conservative summation methods, which sum every convergent series (sequence) to a finite sum, although possibly different from that to which it converges. If a regular summation method is defined by the transformation of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r0808901.png" /> to a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r0808902.png" /> by means of an infinite matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r0808903.png" />:
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{{MSC|40C}}
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{{TEX|done}}
  
<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/r/r080/r080890/r0808904.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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Regular summation methods are those methods for summing series (sequences) that sum every convergent series (sequence) to the same sum as that to which it converges. Regular summation methods are a special case of conservative summation methods, which sum every convergent series (sequence) to a finite sum, although possibly different from that to which it converges. If a regular summation method is defined by the transformation of a sequence $(s_n)$ to a sequence $(\sigma_n)$ by means of an infinite matrix $[a_{nk}]$:
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\begin{equation}
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\label{eq1}
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\sigma_n = \sum_{k=1}^{\infty} a_{nk}s_k, \quad n=1,2,\ldots
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\end{equation}
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(see [[Matrix summation method]]), then the transformation \ref{eq1} and the matrix of this transformation, $[a_{nk}]$, are called regular.
  
(see [[Matrix summation method|Matrix summation method]]), then the transformation (*) and the matrix of this transformation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r0808905.png" />, are called regular.
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Many of the most common summation methods are regular. This applies to the [[Cesàro summation methods]] $(C,k)$ for $k \geq 0$, the [[Hölder summation methods]] and the [[Abel summation method]], among others. There are non-regular summation methods, such as the Cesàro summation method $(C,k)$ for $k<0$, and the [[Riemann summation method]].
  
Many of the most common summation methods are regular. This applies to the [[Cesàro summation methods|Cesàro summation methods]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r0808906.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r0808907.png" />, the [[Hölder summation methods|Hölder summation methods]] and the [[Abel summation method|Abel summation method]], among others. There are non-regular summation methods, such as the Cesàro summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r0808908.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r0808909.png" />, and the [[Riemann summation method|Riemann summation method]].
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A summation method is called completely regular if it is regular and if it sums every series (sequence) with real terms converging to $\infty$ (or $-\infty$) to $\infty$ (respectively, $-\infty$). A regular summation method defined by a positive matrix is completely regular (see also [[Regularity criteria]]).
  
A summation method is called completely regular if it is regular and if it sums every series (sequence) with real terms converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r08089010.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r08089011.png" />) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r08089012.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080890/r08089013.png" />). A regular summation method defined by a positive matrix is completely regular (see also [[Regularity criteria|Regularity criteria]]).
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====References====  
  
====References====
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{|
<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><TR><TD valign="top">[3]</TD> <TD valign="top"G.F. Kangro,   "Theory of summability of sequences and series" ''J. Soviet Math.'' , '''5''' : 1 (1976) pp. 1–45 ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''12''' (1974) pp. 5–70</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Baron,  "Introduction to theory of summation of series" , Tallin  (1977)  (In Russian)</TD></TR></table>
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|valign="top"|{{Ref|Ba}}||valign="top"| S. Baron, "Introduction to theory of summation of series", Tallin (1977) (In Russian)
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|valign="top"|{{Ref|Co}}||valign="top"| R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950)
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|valign="top"|{{Ref|Ha}}||valign="top"| G.H. Hardy, "Divergent series", Clarendon Press (1949)
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|valign="top"|{{Ref|Ka}}||valign="top"| G.F. Kangro, "Theory of summability of sequences and series" ''J. Soviet Math.'', '''5''' : 1 (1976) pp. 1–45 ''Itogi Nauk. i Tekhn. Mat. Anal.'', '''12''' (1974) pp. 5–70
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|-
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|}

Latest revision as of 14:13, 8 May 2012

permanent summation methods

2010 Mathematics Subject Classification: Primary: 40C [MSN][ZBL]

Regular summation methods are those methods for summing series (sequences) that sum every convergent series (sequence) to the same sum as that to which it converges. Regular summation methods are a special case of conservative summation methods, which sum every convergent series (sequence) to a finite sum, although possibly different from that to which it converges. If a regular summation method is defined by the transformation of a sequence $(s_n)$ to a sequence $(\sigma_n)$ by means of an infinite matrix $[a_{nk}]$: \begin{equation} \label{eq1} \sigma_n = \sum_{k=1}^{\infty} a_{nk}s_k, \quad n=1,2,\ldots \end{equation} (see Matrix summation method), then the transformation \ref{eq1} and the matrix of this transformation, $[a_{nk}]$, are called regular.

Many of the most common summation methods are regular. This applies to the Cesàro summation methods $(C,k)$ for $k \geq 0$, the Hölder summation methods and the Abel summation method, among others. There are non-regular summation methods, such as the Cesàro summation method $(C,k)$ for $k<0$, and the Riemann summation method.

A summation method is called completely regular if it is regular and if it sums every series (sequence) with real terms converging to $\infty$ (or $-\infty$) to $\infty$ (respectively, $-\infty$). A regular summation method defined by a positive matrix is completely regular (see also Regularity criteria).

References

[Ba] S. Baron, "Introduction to theory of summation of series", Tallin (1977) (In Russian)
[Co] R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950)
[Ha] G.H. Hardy, "Divergent series", Clarendon Press (1949)
[Ka] G.F. Kangro, "Theory of summability of sequences and series" J. Soviet Math., 5 : 1 (1976) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal., 12 (1974) pp. 5–70
How to Cite This Entry:
Regular summation methods. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Regular_summation_methods&oldid=26222
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article