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''summation by parts''
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{{MSC|40A05}}
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{{TEX|done}}
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''summation by parts, Abel's lemma''
  
A transformation
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A discrete analog of [[Integration by parts|integration by parts]], introduced by N.H. Abel in {{Cite|Ab}}. If $a_1, \ldots, a_N$, $b_1, \ldots, b_N$, are given complex numbers and we set
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\[
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B_n = \sum_{i\leq n} b_i
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\]
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then the summation by parts is the identity
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\[
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\sum_{k=1}^N a_k b_k = a_N B_N - \sum_{k=1}^{N-1} B_k (a_{k+1} - a_k)\, .
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\]
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Observe that, if $B_0$ is arbitrarily chosen and we modify the definition of $B_n$ by
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\[
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B_n = B_0 + \sum_{i\leq n} b_i
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\]
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then the identity becomes
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\[
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\sum_{k=1}^N = a_N B_N - a_1 B_0 - \sum_{k=1}^{N-1} B_k (a_{k+1} - a_k)\, ,
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\]
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in analogy with the arbitrarity of an additive constant in the [[Primitive function|primitive]] of a function.
  
<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/a/a010/a010190/a0101901.png" /></td> </tr></table>
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If $a_n\to 0$ and $\{B_n\}$ is a bounded sequence, the Abel transformation shows that $\sum_k a_k b_k$ converges if and only if $\sum_k B_k (a_{k+1}-a_k)$ converges, in which case it yields the formula
 
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\[
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101902.png" /> are given, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101903.png" /> is arbitrarily selected, and
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\sum_{k=1}^\infty a_k b_k = \sum_{k=1}^\infty B_k (a_k - a_{k+1}) - a_1 B_0\, .
 
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\]
<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/a/a010/a010190/a0101904.png" /></td> </tr></table>
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This fact can be used to prove several very useful criteria of convergence of series of numbers and functions (cf. [[Abel criterion]]). The Abel transformation of a series often yields a series with an identical sum, but with a better convergence. It is also regularly used to obtain certain estimates (cf. [[Abel inequality|Abel inequality]]), in particular, for investigations on the rate of convergence of a series.  
 
 
The Abel transformation is the discrete analogue of the formula for [[Integration by parts|integration by parts]].
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101905.png" /> and if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101906.png" /> is bounded, then the Abel transformation can be applied to 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/a/a010/a010190/a0101907.png" /></td> </tr></table>
 
 
 
The Abel transformation is used to prove several criteria of convergence of series of numbers and functions (cf. [[Abel criterion|Abel criterion]]). The Abel transformation of a series often yields a series with an identical sum, but with a better convergence. It is also regularly used to obtain certain estimates (cf. [[Abel inequality|Abel inequality]]), in particular, for investigations on the rate of convergence of a series. It was introduced by N.H. Abel [[#References|[1]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.H. Abel,  "Untersuchungen über die Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010190/a0101908.png" />"  ''J. Reine Angew. Math.'' , '''1'''  (1826)  pp. 311–339</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , '''1–2''' , Cambridge Univ. Press  (1952)</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Ab}}|| N.H. Abel,  "Untersuchungen über die Reihe $1+ \frac{m}{x} + \frac{m\cdot (m-1)}{2\cdot 1} x^2 + \frac{m\cdot (m-1)\cdot (m-2)}{3\cdot 2\cdot 1} x^3 + \ldots$ u.s.w.", ''J. Reine Angew. Math.'' , '''1'''  (1826)  pp. 311–339
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|-
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|valign="top"|{{Ref|Ca}}||  H. Cartan, "Elementary Theory of Analytic Functions of One or Several  Complex Variable", Dover (1995).
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|-
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|valign="top"|{{Ref|Ma}}|| A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)
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|-
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|valign="top"|{{Ref|WW}}||  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , '''1–2''' , Cambridge Univ. Press  (1952)
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|}

Latest revision as of 13:06, 10 December 2013

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL] summation by parts, Abel's lemma

A discrete analog of integration by parts, introduced by N.H. Abel in [Ab]. If $a_1, \ldots, a_N$, $b_1, \ldots, b_N$, are given complex numbers and we set \[ B_n = \sum_{i\leq n} b_i \] then the summation by parts is the identity \[ \sum_{k=1}^N a_k b_k = a_N B_N - \sum_{k=1}^{N-1} B_k (a_{k+1} - a_k)\, . \] Observe that, if $B_0$ is arbitrarily chosen and we modify the definition of $B_n$ by \[ B_n = B_0 + \sum_{i\leq n} b_i \] then the identity becomes \[ \sum_{k=1}^N = a_N B_N - a_1 B_0 - \sum_{k=1}^{N-1} B_k (a_{k+1} - a_k)\, , \] in analogy with the arbitrarity of an additive constant in the primitive of a function.

If $a_n\to 0$ and $\{B_n\}$ is a bounded sequence, the Abel transformation shows that $\sum_k a_k b_k$ converges if and only if $\sum_k B_k (a_{k+1}-a_k)$ converges, in which case it yields the formula \[ \sum_{k=1}^\infty a_k b_k = \sum_{k=1}^\infty B_k (a_k - a_{k+1}) - a_1 B_0\, . \] This fact can be used to prove several very useful criteria of convergence of series of numbers and functions (cf. Abel criterion). The Abel transformation of a series often yields a series with an identical sum, but with a better convergence. It is also regularly used to obtain certain estimates (cf. Abel inequality), in particular, for investigations on the rate of convergence of a series.

References

[Ab] N.H. Abel, "Untersuchungen über die Reihe $1+ \frac{m}{x} + \frac{m\cdot (m-1)}{2\cdot 1} x^2 + \frac{m\cdot (m-1)\cdot (m-2)}{3\cdot 2\cdot 1} x^3 + \ldots$ u.s.w.", J. Reine Angew. Math. , 1 (1826) pp. 311–339
[Ca] H. Cartan, "Elementary Theory of Analytic Functions of One or Several Complex Variable", Dover (1995).
[Ma] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[WW] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , 1–2 , Cambridge Univ. Press (1952)
How to Cite This Entry:
Abel transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_transformation&oldid=30927
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article