View source for Row-finite summation method

A [[Matrix summation method|matrix summation method]] determined by a row-finite matrix, that is, a matrix in which each row has only finitely many non-zero entries. An important special case of a row-finite summation method is a [[Triangular summation method|triangular summation method]].

For every regular matrix summation method for sequences (cf. [[Regular summation methods|Regular summation methods]]), a row-finite summation method that is equivalent and compatible (see [[Inclusion of summation methods|Inclusion of summation methods]]; [[Compatibility of summation methods|Compatibility of summation methods]]) with it on the set of all bounded sequences can be constructed (see [[#References|[3]]]). However, there exist regular matrix summation methods for which there is no equivalent row-finite summation method on the set of all sequences (see [[#References|[4]]] for an example).

====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">  A.L. Brudno,   "Summation of bounded sequences by matrices"  ''Mat. Sb.'' , '''16'''  (1945)  pp. 191–247  (In Russian)  (English abstract)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Erdös,   G. Piranian,   "Convergence fields of row-finite and row-infinite Toeplitz transformations"  ''Proc. Amer. Math. Soc.'' , '''1'''  (1950)  pp. 397–401</TD></TR></table>