# Triangular summation method

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2010 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.

How to Cite This Entry:
Triangular summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangular_summation_method&oldid=25646
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article