Difference between revisions of "Triangular summation method"
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A [[Matrix summation method|matrix summation method]] defined by a [[Triangular matrix|triangular matrix]] | A [[Matrix summation method|matrix summation method]] defined by a [[Triangular matrix|triangular matrix]] | ||
| − | + | $$A=\| a_{nk}\|,\quad n,k=1,2,\ldots,$$ | |
| − | that is, by a matrix for which | + | that is, by a matrix for which $a_{nk}=0$ for $k>n$. A triangulation summation method is a special case of a [[Row-finite summation method|row-finite summation method]]. A triangular matrix $A$ is called normal if $a_{nn}\neq0$ for all $n$. The transformation |
| − | + | $$\sigma_n=\sum_{k=1}^na_{nk}s_k$$ | |
| − | realized by a normal triangular matrix | + | realized by a normal triangular matrix $A$ has an inverse: |
| − | + | $$s_n=\sum_{k=1}^na_{nk}^{-1}\sigma_k,$$ | |
| − | where | + | where $A^{-1}=\| a_{nk}^{-1}\|$ is the inverse of $A$. This fact simplifies the proof of a number of theorems for matrix summation methods determined by normal triangular matrices. Related to the triangular summation methods are, e.g., the [[Cesàro summation methods|Cesàro summation methods]] and the [[Voronoi summation method|Voronoi summation method]]. |
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Ba}}||valign="top"| S.A. Baron, "Introduction to the theory of summability of series", Tartu (1966) (In Russian) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Co}}||valign="top"| R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ha}}||valign="top"| G.H. Hardy, "Divergent series", Clarendon Press (1949) | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 10:25, 28 April 2012
2020 Mathematics Subject Classification: Primary: 40C05 [MSN][ZBL]
A matrix summation method defined by a triangular matrix
$$A=\| a_{nk}\|,\quad n,k=1,2,\ldots,$$
that is, by a matrix for which $a_{nk}=0$ for $k>n$. A triangulation summation method is a special case of a row-finite summation method. A triangular matrix $A$ is called normal if $a_{nn}\neq0$ for all $n$. The transformation
$$\sigma_n=\sum_{k=1}^na_{nk}s_k$$
realized by a normal triangular matrix $A$ has an inverse:
$$s_n=\sum_{k=1}^na_{nk}^{-1}\sigma_k,$$
where $A^{-1}=\| a_{nk}^{-1}\|$ is the inverse of $A$. This fact simplifies the proof of a number of theorems for matrix summation methods determined by normal triangular matrices. Related to the triangular summation methods are, e.g., the Cesàro summation methods and the Voronoi summation method.
References
| [Ba] | S.A. Baron, "Introduction to the theory of summability of series", Tartu (1966) (In Russian) |
| [Co] | R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950) |
| [Ha] | G.H. Hardy, "Divergent series", Clarendon Press (1949) |
How to Cite This Entry:
Triangular summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangular_summation_method&oldid=15775
Triangular summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangular_summation_method&oldid=15775
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article