Difference between revisions of "Regular summation methods"
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''permanent summation methods'' | ''permanent summation methods'' | ||
− | + | {{MSC|40C}} | |
+ | {{TEX|done}} | ||
− | + | 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==== | |
− | + | {| | |
− | + | |- | |
+ | |valign="top"|{{Ref|Ba}}||valign="top"| S. Baron, "Introduction to theory of summation of series", Tallin (1977) (In Russian) | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Co}}||valign="top"| R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950) | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Ha}}||valign="top"| G.H. Hardy, "Divergent series", Clarendon Press (1949) | ||
+ | |- | ||
+ | |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 | ||
+ | |- | ||
+ | |} |
Latest revision as of 13:13, 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 |
Regular summation methods. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Regular_summation_methods&oldid=26222