Hausdorff summation method
A summation method for series of numbers or functions, introduced by F. Hausdorff ; 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
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 .
|||F. Hausdorff, "Summationsmethoden und Momentfolgen I, II" Math. Z. , 9 (1921) pp. 74–109; 280–299|
|||G.H. Hardy, "Divergent series" , Clarendon Press (1949)|
Hausdorff summation method. I.I. Volkov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hausdorff_summation_method&oldid=12912