# 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 <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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− | 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> | |

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− | 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 <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: |

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− | 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> | |

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− | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080910/r0809108.png" />-matrix. |

− | 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 <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. |

− | or a transformation of a series into a function by means of a semi-continuous matrix | ||

− | 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> | ||

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====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 <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" />. |

− | with | ||

− | for all | ||

− | with |

## Revision as of 14:53, 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 , the conditions

(1) |

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 , necessary and sufficient conditions for regularity are as follows:

(2) |

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 -matrix.

For a semi-continuous summation method, defined by a transformation of a sequence into a function by means of a semi-continuous matrix or a transformation of a series into a function by means of a semi-continuous matrix , 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 with for all with .

**How to Cite This Entry:**

Regularity criteria.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Regularity_criteria&oldid=48488