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Difference between revisions of "Inclusion of summation methods"

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An inclusion of the summability fields (cf. [[Summability field|Summability field]]) corresponding to these methods. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i0504401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i0504402.png" /> be two [[Summation methods|summation methods]] defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i0504403.png" /> of series (or sequences); let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i0504404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i0504405.png" /> be their summability fields and suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i0504406.png" />; one then says that method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i0504407.png" /> includes method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i0504408.png" />, which is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i0504409.png" />. Methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044011.png" /> are said to be equipotent, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044012.png" />, if each of them includes the other. Equipotent methods have the same summability field. Method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044013.png" /> is said to be stronger than method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044014.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044015.png" /> includes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044016.png" /> but is not equipotent with it. If the summability field of the method is identical with the set of all convergent series, the method is said to be equipotent with convergence. An inclusion of summation methods is sometimes considered not on the entire set of their definition, but only on some of its subsets.
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An inclusion of the summability fields (cf. [[Summability field|Summability field]]) corresponding to these methods. Let $A$ and $B$ be two [[Summation methods|summation methods]] defined on a set $M$ of series (or sequences); let $A^*$ and $B^*$ be their summability fields and suppose $A^*\subset B^*$ one then says that method $B$ includes method $A$, which is denoted by the symbol $A\subset B$. Methods $A$ and $B$ are said to be equipotent, denoted by $A=B$, if each of them includes the other. Equipotent methods have the same summability field. Method $B$ is said to be stronger than method $A$ if $B$ includes $A$ but is not equipotent with it. If the summability field of the method is identical with the set of all convergent series, the method is said to be equipotent with convergence. An inclusion of summation methods is sometimes considered not on the entire set of their definition, but only on some of its subsets.
  
For the [[Cesàro summation methods|Cesàro summation methods]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044017.png" /> the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044018.png" /> is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044019.png" />; the [[Abel summation method|Abel summation method]] is stronger than all Cesàro methods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044020.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044021.png" />; the [[Riesz summation method|Riesz summation method]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044022.png" /> is equipotent with the Cesàro summation method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044024.png" />); the Abel summation method is equipotent with convergence on the set of series whose terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044025.png" /> satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044026.png" />. In these examples the summation methods are compatible as well (cf. [[Compatibility of summation methods|Compatibility of summation methods]]), even though, in general, an inclusion of summation methods does not rest on the assumption of their being compatible. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044028.png" /> are regular matrix methods (cf. [[Regular summation methods|Regular summation methods]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044029.png" /> on the set of bounded sequences, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044031.png" /> are compatible on this set (the Mazur–Orlicz–Brudno theorem). The compatibility requirement is imposed in the very definition of inclusion in certain textbooks.
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For the [[Cesàro summation methods|Cesàro summation methods]] $(C,k)$ the inclusion $(C,k_1)\subset(C,k_2)$ is valid for $k_2\geq k_1>-1$; the [[Abel summation method|Abel summation method]] is stronger than all Cesàro methods $(C,k)$ for $k>-1$; the [[Riesz summation method|Riesz summation method]] $(R,n,k)$ is equipotent with the Cesàro summation method $(C,k)$ ($k\geq0$); the Abel summation method is equipotent with convergence on the set of series whose terms $a_n$ satisfy the condition $a_n=O(1/n)$. In these examples the summation methods are compatible as well (cf. [[Compatibility of summation methods|Compatibility of summation methods]]), even though, in general, an inclusion of summation methods does not rest on the assumption of their being compatible. However, if $A$ and $B$ are regular matrix methods (cf. [[Regular summation methods|Regular summation methods]]) and $A\subset B$ on the set of bounded sequences, then $A$ and $B$ are compatible on this set (the Mazur–Orlicz–Brudno theorem). The compatibility requirement is imposed in the very definition of inclusion in certain textbooks.
  
An inclusion of summation methods defined on a set of series with real terms is said to be complete if the inclusion of their summability fields is preserved after these fields have been completed with series summable to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044033.png" />. Thus, the Hölder summation method (cf. [[Hölder summation methods|Hölder summation methods]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044034.png" /> completely includes the Cesàro method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050440/i05044035.png" />.
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An inclusion of summation methods defined on a set of series with real terms is said to be complete if the inclusion of their summability fields is preserved after these fields have been completed with series summable to $+\infty$ and $-\infty$. Thus, the Hölder summation method (cf. [[Hölder summation methods|Hölder summation methods]]) $(H,k)$ completely includes the Cesàro method $(C,k)$.
  
 
An inclusion of summation methods for special types of summability (e.g. absolute summability, strong summability, etc.) is defined in a similar manner.
 
An inclusion of summation methods for special types of summability (e.g. absolute summability, strong summability, etc.) is defined in a similar manner.

Latest revision as of 22:16, 7 July 2014

An inclusion of the summability fields (cf. Summability field) corresponding to these methods. Let $A$ and $B$ be two summation methods defined on a set $M$ of series (or sequences); let $A^*$ and $B^*$ be their summability fields and suppose $A^*\subset B^*$ one then says that method $B$ includes method $A$, which is denoted by the symbol $A\subset B$. Methods $A$ and $B$ are said to be equipotent, denoted by $A=B$, if each of them includes the other. Equipotent methods have the same summability field. Method $B$ is said to be stronger than method $A$ if $B$ includes $A$ but is not equipotent with it. If the summability field of the method is identical with the set of all convergent series, the method is said to be equipotent with convergence. An inclusion of summation methods is sometimes considered not on the entire set of their definition, but only on some of its subsets.

For the Cesàro summation methods $(C,k)$ the inclusion $(C,k_1)\subset(C,k_2)$ is valid for $k_2\geq k_1>-1$; the Abel summation method is stronger than all Cesàro methods $(C,k)$ for $k>-1$; the Riesz summation method $(R,n,k)$ is equipotent with the Cesàro summation method $(C,k)$ ($k\geq0$); the Abel summation method is equipotent with convergence on the set of series whose terms $a_n$ satisfy the condition $a_n=O(1/n)$. In these examples the summation methods are compatible as well (cf. Compatibility of summation methods), even though, in general, an inclusion of summation methods does not rest on the assumption of their being compatible. However, if $A$ and $B$ are regular matrix methods (cf. Regular summation methods) and $A\subset B$ on the set of bounded sequences, then $A$ and $B$ are compatible on this set (the Mazur–Orlicz–Brudno theorem). The compatibility requirement is imposed in the very definition of inclusion in certain textbooks.

An inclusion of summation methods defined on a set of series with real terms is said to be complete if the inclusion of their summability fields is preserved after these fields have been completed with series summable to $+\infty$ and $-\infty$. Thus, the Hölder summation method (cf. Hölder summation methods) $(H,k)$ completely includes the Cesàro method $(C,k)$.

An inclusion of summation methods for special types of summability (e.g. absolute summability, strong summability, etc.) is defined in a similar manner.

References

[1] G.H. Hardy, "Divergent series" , Oxford Univ. Press (1949)
[2] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[3] G.P. Kangro, "Theory of summability of sequences and series" J. Soviet Math. , 5 : 1 (1976) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 5–70
[4] S. Mazur, W. Orlicz, "Sur les méthodes linéaires de sommation" C.R. Acad. Sci. Paris Sér. I Math. , 196 (1933) pp. 32–34
[5] A.L. Brudno, "Summability of bounded sequences of matrices" Mat. Sb. , 16 (58) : 2 (1945) pp. 191–247 (In Russian) (English abstract)
[6] S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian)
How to Cite This Entry:
Inclusion of summation methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inclusion_of_summation_methods&oldid=15184
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article