Regularity criteria

for summation methods

Conditions for the regularity of summation methods.

For a matrix summation method defined by a transformation of a sequence into a sequence by means of a matrix , the conditions

(1)

are necessary and sufficient for regularity. For the matrix summation method defined by a transformation of a series into a sequence by means of a matrix , necessary and sufficient conditions for regularity are as follows:

(2)

The conditions (1) were originally established by O. Toeplitz [1] for triangular summation methods, and were then extended by H. Steinhaus [2] to arbitrary matrix summation methods. In connection with this, a matrix satisfying conditions (1) is sometimes called a Toeplitz matrix or a -matrix.

For a semi-continuous summation method, defined by a transformation of a sequence into a function by means of a semi-continuous matrix or a transformation of a series into a function by means of a semi-continuous matrix , there are regularity criteria analogous to conditions (1) and (2), respectively.

A regular matrix summation method is completely regular if all entries of the transformation matrix are non-negative. This condition is in general not necessary for complete regularity.

References

[1]	O. Toeplitz, Prace Mat. Fiz. , 22 (1911) pp. 113–119
[2]	H. Steinhaus, "Some remarks on the generalization of the concept of limit" , Selected Math. Papers , Polish Acad. Sci. (1985) pp. 88–100
[3]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[4]	R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)

Comments

Cf. also Regular summation methods.

Usually, the phrase Toeplitz matrix refers to a matrix with for all with .