# 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)

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