# 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 $\mathcal{A}$ 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) |

**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