View source for Hausdorff summation method

A summation method for series of numbers or functions, introduced by F. Hausdorff [[#References|[1]]]; it is defined as follows. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h0467401.png" /> is subjected in succession to three linear matrix transformations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h0467402.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h0467403.png" /> is the transformation by means of the triangular matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h0467404.png" />:

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and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h0467406.png" /> is the diagonal transformation by means the diagonal matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h0467407.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h0467408.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h0467409.png" /> is a numerical sequence. The transformation

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674012.png" /> is an arbitrary numerical sequence, is called a general Hausdorff transformation, and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674013.png" /> — a Hausdorff matrix. Written out, a general Hausdorff transformation has the form

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where

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674016.png" /></td> </tr></table>

The series

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with partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674018.png" /> is summable by the Hausdorff method to sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674019.png" /> if

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The field and the regularity of the Hausdorff method depend on the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674022.png" /> is a real sequence, then for the regularity of the method it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674023.png" /> is the difference of two absolutely-monotone sequences and that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674024.png" /></td> </tr></table>

or, in another terminology, necessary and sufficient is that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674025.png" /> are regular moments.

The Hausdorff summation method contains as special cases a number of other well-known summation methods. Thus, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674026.png" /> the Hausdorff method turns into the Euler method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674027.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674028.png" /> into the Hölder method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674029.png" />, and for

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into the Cesàro method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674031.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hausdorff,   "Summationsmethoden und Momentfolgen I, II"  ''Math. Z.'' , '''9'''  (1921)  pp. 74–109; 280–299</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR></table>