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The property of the method consisting in the preservation of summability of a series after adding to or deleting from it a finite number of terms. More precisely, a summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t0938701.png" /> is said to be translative if the summability of the series
+
The property of the method consisting in the preservation of summability of a series after adding to or deleting from it a finite number of terms. More precisely, a summation method $\mathcal{A}$ is said to be translative if the summability 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/t/t093/t093870/t0938702.png" /></td> </tr></table>
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\sum_{k=0}^\infty a_k
 
+
$$
to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t0938703.png" /> implies that the series
+
to the sum $S_1$ implies that 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/t/t093/t093870/t0938704.png" /></td> </tr></table>
+
\sum_{k=1}^\infty a_k
 
+
$$
is summable by the same method to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t0938705.png" />, and conversely. For a summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t0938706.png" /> defined by transformation of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t0938707.png" /> into a sequence or function, the property of translativity consists of the equivalence of the conditions
+
is summable by the same method to the sum $S_1 - a_0$, and conversely. For a summation method $\mathcal{A}$ defined by transformation of the sequence $S_n$ into a sequence or function, the property of translativity consists of the equivalence of the conditions
 
+
$$
<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/t/t093/t093870/t0938708.png" /></td> </tr></table>
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\mathcal{A}\text{-}\lim S_n = S
 
+
$$
 
and
 
and
 +
$$
 +
\mathcal{A}\text{-}\lim S_{n+1} = S
 +
$$
  
<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/t/t093/t093870/t0938709.png" /></td> </tr></table>
+
If the summation method is defined by a regular matrix $(A_{nk})$ (cf. [[Regular summation methods|Regular summation methods]]), then this means that
 
+
$$\label{eq:a1}
If the summation method is defined by a regular matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t09387010.png" /> (cf. [[Regular summation methods|Regular summation methods]]), then this means that
+
\lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_k = S
 
+
$$
<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/t/t093/t093870/t09387011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
 
 
 
 
always implies that
 
always implies that
 +
$$\label{eq:a2}
 +
\lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_{k+1} = S
 +
$$
 +
and conversely. In cases when such an inference only holds in one direction, the method is called right translative if \ref{eq:a1} implies \ref{eq:a2} but the converse is false, or left translative if \ref{eq:a2} implies \ref{eq:a1} but the converse is false.
  
<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/t/t093/t093870/t09387012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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Many widely used summation methods have the property of translativity; for example, the [[Cesàro summation methods]] $(C,k)$ for $k > 0$, the [[Riesz summation method]] $R(n,k)$ for $k>0$ and the [[Abel summation method]] are translative; the [[Borel summation method]] is left translative.
  
and conversely. In cases when such an inference only holds in one direction, the method is called right translative if (1) implies (2) but the converse is false, or left translative if (2) implies (1) but the converse is false.
+
====References====
 +
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  R.G. Cooke,  "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  S.A. Baron,  "Introduction to the theory of summability of series" , Tartu  (1966) (In Russian)</TD></TR>
 +
</table>
  
Many widely used summation methods have the property of translativity; for example, the [[Cesàro summation methods|Cesàro summation methods]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t09387013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t09387014.png" />, the [[Riesz summation method|Riesz summation method]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t09387015.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t09387016.png" /> and the [[Abel summation method|Abel summation method]] are translative; the [[Borel summation method|Borel summation method]] is left translative.
+
{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.G. Cooke,  "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.A. Baron,  "Introduction to the theory of summability of series" , Tartu  (1966)  (In Russian)</TD></TR></table>
 

Latest revision as of 19:29, 23 December 2015

The property of the method consisting in the preservation of summability of a series after adding to or deleting from it a finite number of terms. More precisely, a summation method $\mathcal{A}$ is said to be translative if the summability of the series $$ \sum_{k=0}^\infty a_k $$ to the sum $S_1$ implies that the series $$ \sum_{k=1}^\infty a_k $$ is summable by the same method to the sum $S_1 - a_0$, and conversely. For a summation method $\mathcal{A}$ defined by transformation of the sequence $S_n$ into a sequence or function, the property of translativity consists of the equivalence of the conditions $$ \mathcal{A}\text{-}\lim S_n = S $$ and $$ \mathcal{A}\text{-}\lim S_{n+1} = S $$

If the summation method is defined by a regular matrix $(A_{nk})$ (cf. Regular summation methods), then this means that $$\label{eq:a1} \lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_k = S $$ always implies that $$\label{eq:a2} \lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_{k+1} = S $$ and conversely. In cases when such an inference only holds in one direction, the method is called right translative if \ref{eq:a1} implies \ref{eq:a2} but the converse is false, or left translative if \ref{eq:a2} implies \ref{eq:a1} but the converse is false.

Many widely used summation methods have the property of translativity; for example, the Cesàro summation methods $(C,k)$ for $k > 0$, the Riesz summation method $R(n,k)$ for $k>0$ and the Abel summation method are translative; the Borel summation method is left translative.

References

[1] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[2] S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian)
How to Cite This Entry:
Translativity of a summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Translativity_of_a_summation_method&oldid=17808
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article