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''for summation methods''
 
''for summation methods''
  
 
Conditions for the regularity of [[Summation methods|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r0809101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r0809102.png" /> the conditions
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r0809103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\left .
 +
\begin{array}{l}
 +
\textrm{ 1)  } \  \sum _ { k= } 1 ^  \infty  | a _ {nk} |  \leq  M ; \\
 +
\textrm{ 2)  } \  \lim\limits _ {n \rightarrow \infty }  a _ {nk}  = 0 ; \\
 +
\textrm{ 3)  } \  \lim\limits _ {n \rightarrow \infty }  \sum _ { k= } 1 ^  \infty 
 +
a _ {nk}  = 1 ,  \\
 +
\end{array}
 +
\right \}
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r0809104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r0809105.png" /> necessary and sufficient conditions for regularity are as follows:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r0809106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\left .
 +
\begin{array}{l}
 +
\textrm{ 1)  } \  \sum _ { k= } 1 ^  \infty  | g _ {n,k} - g _ {n,k-} 1 |  \leq  M ; \\
 +
\textrm{ 2)  } \  \lim\limits _ {n \rightarrow \infty }  g _ {nk}  = 1 . \\
 +
\end{array}
 +
\right \}
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r0809108.png" />-matrix.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r0809109.png" /> or a transformation of a series into a function by means of a semi-continuous matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r08091010.png" />, there are regularity criteria analogous to conditions (1) and (2), respectively.
+
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.
 
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.
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====References====
 
====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>
 
<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====
 
====Comments====
 
Cf. also [[Regular summation methods|Regular summation methods]].
 
Cf. also [[Regular summation methods|Regular summation methods]].
  
Usually, the phrase [[Toeplitz matrix|Toeplitz matrix]] refers to a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r08091011.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r08091012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r08091013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r08091014.png" />.
+
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 $.

Revision as of 14:55, 7 June 2020


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 $.

How to Cite This Entry:
Regularity criteria. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regularity_criteria&oldid=49557
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article