Difference between revisions of "Translativity of a summation method"
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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 | + | 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 is said to be translative if the summability of the series |
− | + | $$ | |
− | + | \sum_{k=0}^\infty a_k | |
− | + | $$ | |
− | to the sum | + | to the sum S_1 implies that the series |
− | + | $$ | |
− | + | \sum_{k=1}^\infty a_k | |
− | + | $$ | |
− | is summable by the same method to the sum | + | 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 | 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|Regular summation methods]]), then this means that | |
− | + | $$\label{eq:a1} | |
− | If the summation method is defined by a regular matrix | + | \lim_{n\rightarrow\infty} \sum_{k=0}^\infty A_{nk} S_k = S |
− | + | $$ | |
− | |||
− | |||
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. | ||
− | + | 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 | + | ====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> | ||
− | + | {{TEX|done}} | |
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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) |
Translativity of a summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Translativity_of_a_summation_method&oldid=17808