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 is said to be translative if the summability of the series

to the sum implies that the series

is summable by the same method to the sum , and conversely. For a summation method defined by transformation of the sequence into a sequence or function, the property of translativity consists of the equivalence of the conditions

and

If the summation method is defined by a regular matrix (cf. Regular summation methods), then this means that

(1)

always implies that

(2)

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 for , the Riesz summation method for 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)