Hausdorff summation method

A summation method for series of numbers or functions, introduced by F. Hausdorff [1]; it is defined as follows. A sequence is subjected in succession to three linear matrix transformations:

where is the transformation by means of the triangular matrix :

and is the diagonal transformation by means the diagonal matrix :

where is a numerical sequence. The transformation

where , is an arbitrary numerical sequence, is called a general Hausdorff transformation, and the matrix — a Hausdorff matrix. Written out, a general Hausdorff transformation has the form

where

The series

with partial sums is summable by the Hausdorff method to sum if

The field and the regularity of the Hausdorff method depend on the sequence . If is a real sequence, then for the regularity of the method it is necessary and sufficient that is the difference of two absolutely-monotone sequences and that

or, in another terminology, necessary and sufficient is that the are regular moments.

The Hausdorff summation method contains as special cases a number of other well-known summation methods. Thus, for the Hausdorff method turns into the Euler method , for into the Hölder method , and for

into the Cesàro method .

References

[1]	F. Hausdorff, "Summationsmethoden und Momentfolgen I, II" Math. Z. , 9 (1921) pp. 74–109; 280–299
[2]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)