View source for Translativity of a summation method

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

<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>

to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093870/t0938703.png" /> 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>

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

<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>

and

<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 <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

<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

<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>

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.

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.

====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>