# Difference between revisions of "Regularity criteria"

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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 | + | 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 | + | 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 | + | 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 | + | 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. | ||

Line 19: | Line 54: | ||

====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 | + | 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 $. |

## Latest revision as of 16:17, 22 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=17191