# 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  for triangular summation methods, and were then extended by H. Steinhaus  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.

How to Cite This Entry:
Regularity criteria. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regularity_criteria&oldid=49805
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article