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Row-finite summation method

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A 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.

For every regular matrix summation method for sequences (cf. Regular summation methods), a row-finite summation method that is equivalent and compatible (see Inclusion of summation methods; Compatibility of summation methods) with it on the set of all bounded sequences can be constructed (see [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 [4] for an example).

References

[1] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[2] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[3] A.L. Brudno, "Summation of bounded sequences by matrices" Mat. Sb. , 16 (1945) pp. 191–247 (In Russian) (English abstract)
[4] P. Erdös, G. Piranian, "Convergence fields of row-finite and row-infinite Toeplitz transformations" Proc. Amer. Math. Soc. , 1 (1950) pp. 397–401
How to Cite This Entry:
Row-finite summation method. I.I. Volkov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Row-finite_summation_method&oldid=17270
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098