Regularity criteria

for summation methods

Conditions for the regularity of summation methods.

For a matrix summation method defined by a transformation of a sequence into a sequence by means of a matrix $ \| a _ {nk} \| $, $ n , k = 1 , 2 \dots $ the conditions

$$ \tag{1 } \left . \begin{array}{l} \textrm{ 1) } \ \sum _ { k=1 } ^ \infty | a _ {nk} | \leq M ; \\ \textrm{ 2) } \ \lim\limits _ {n \rightarrow \infty } a _ {nk} = 0 ; \\ \textrm{ 3) } \ \lim\limits _ {n \rightarrow \infty } \sum _ { k=1 } ^ \infty a _ {nk} = 1 , \\ \end{array} \right \} $$

are necessary and sufficient for regularity. For the matrix summation method defined by a transformation of a series into a sequence by means of a matrix $ \| g _ {nk} \| $, $ n , k = 1 , 2 \dots $ necessary and sufficient conditions for regularity are as follows:

$$ \tag{2 } \left . \begin{array}{l} \textrm{ 1) } \ \sum _ { k=1 } ^ \infty | g _ {n,k} - g _ {n,k-1} | \leq M ; \\ \textrm{ 2) } \ \lim\limits _ {n \rightarrow \infty } g _ {nk} = 1 . \\ \end{array} \right \} $$

The conditions (1) were originally established by O. Toeplitz [1] for triangular summation methods, and were then extended by H. Steinhaus [2] to arbitrary matrix summation methods. In connection with this, a matrix satisfying conditions (1) is sometimes called a Toeplitz matrix or a $ T $- matrix.

For a semi-continuous summation method, defined by a transformation of a sequence into a function by means of a semi-continuous matrix $ \| a _ {k} ( \omega ) \| $ or a transformation of a series into a function by means of a semi-continuous matrix $ \| g _ {k} ( \omega ) \| $, there are regularity criteria analogous to conditions (1) and (2), respectively.

A regular matrix summation method is completely regular if all entries of the transformation matrix are non-negative. This condition is in general not necessary for complete regularity.

References

[1]	O. Toeplitz, Prace Mat. Fiz. , 22 (1911) pp. 113–119
[2]	H. Steinhaus, "Some remarks on the generalization of the concept of limit" , Selected Math. Papers , Polish Acad. Sci. (1985) pp. 88–100
[3]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[4]	R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)

Comments

Cf. also Regular summation methods.

Usually, the phrase Toeplitz matrix refers to a matrix $ ( a _ {ij} ) $ with $ a _ {ij} = a _ {kl} $ for all $ i, j, k, l $ with $ i- j= k- l $.