Difference between revisions of "Triangular summation method"
From Encyclopedia of Mathematics
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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]] | ||
| − | + | $$A=\| a_{nk}\|,\,\,\,n,k=1,2,...,$$ | |
| − | 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==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.H. Hardy, "Divergent series" , Clarendon Press (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.H. Hardy, "Divergent series" , Clarendon Press (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian)</TD></TR></table> | ||
Revision as of 20:48, 19 April 2012
A matrix summation method defined by a triangular matrix
$$A=\| a_{nk}\|,\,\,\,n,k=1,2,...,$$
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
| [1] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) |
| [2] | R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950) |
| [3] | S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian) |
How to Cite This Entry:
Triangular summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangular_summation_method&oldid=24809
Triangular summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangular_summation_method&oldid=24809
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article