Triangular summation method

A matrix summation method defined by a triangular matrix

$$A=\| a_{nk}\|,\,\,\,n,k=1,2,\ldots,$$

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

[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)