View source for Regularity criteria

<!--
r0809101.png
$#A+1 = 13 n = 0
$#C+1 = 13 : ~/encyclopedia/old_files/data/R080/R.0800910 Regularity criteria
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''for summation methods''

Conditions for the regularity of [[Summation methods|summation methods]].

For a [[Matrix summation method|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 [[#References|[1]]] for triangular summation methods, and were then extended by H. Steinhaus [[#References|[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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Toeplitz,   ''Prace Mat. Fiz.'' , '''22'''  (1911)  pp. 113–119</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Steinhaus,   "Some remarks on the generalization of the concept of limit" , ''Selected Math. Papers'' , Polish Acad. Sci.  (1985)  pp. 88–100</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR></table>

====Comments====
Cf. also [[Regular summation methods|Regular summation methods]].

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