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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:
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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/h0467402.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
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" />:
+
A summation method for series of numbers or functions, introduced by F. Hausdorff [[#References|[1]]]; it is defined as follows. A sequence  $  s = \{ s _ {n} \} $
 +
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/h0467405.png" /></td> </tr></table>
+
$$
 +
= \delta s,\ \
 +
\tau  = \mu t,\ \
 +
\sigma  = \delta t,
 +
$$
  
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" />:
+
where  $  \delta $
 +
is the transformation by means of the triangular matrix $  \{ \delta _ {nk} \} $:
  
<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>
+
$$
 +
\delta _ {nk}  = \
 +
\left \{
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h0467409.png" /> is a numerical sequence. The transformation
+
\begin{array}{ll}
 +
(- 1)  ^ {k} \left ( \begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right ) ,  & k \leq  n,  \\
 +
0 ,  & k > n;  \\
 +
\end{array}
  
<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/h04674010.png" /></td> </tr></table>
+
$$
  
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
+
and  $  \mu $
 +
is the diagonal transformation by means the diagonal matrix $  \| \mu _ {nk} \| $:
  
<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/h04674014.png" /></td> </tr></table>
+
$$
 +
\mu _ {nk}  = \
 +
\left \{
 +
 
 +
\begin{array}{ll}
 +
\mu _ {n} ,  & k = n,  \\
 +
0,  & k \neq n,  \\
 +
\end{array}
 +
 
 +
\right .$$
 +
 
 +
where  $  \mu _ {n} $
 +
is a numerical sequence. The transformation
 +
 
 +
$$
 +
\sigma  = \lambda s,
 +
$$
 +
 
 +
where  $  \lambda = \delta \mu \delta $,
 +
$  \{ \mu _ {n} \} $
 +
is an arbitrary numerical sequence, is called a general Hausdorff transformation, and the matrix  $  \delta \mu \delta $—
 +
a Hausdorff matrix. Written out, a general Hausdorff transformation has the form
 +
 
 +
$$
 +
\sigma _ {n}  = \
 +
\sum _ {k = 0 } ^ { n }
 +
\lambda _ {nk} s _ {k} ,
 +
$$
  
 
where
 
where
  
<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/h04674015.png" /></td> </tr></table>
+
$$
 +
\lambda _ {nk}  = \
 +
\left \{
 +
 
 +
\begin{array}{ll}
 +
\left ( \begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right )
 +
\Delta ^ {n - k } \mu _ {k} ,  & k \leq  n,  \\
 +
0,  & k > n;  \\
 +
\end{array}
 +
 
 +
$$
  
<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>
+
$$
 +
\Delta \mu _ {k}  = \mu _ {k} - \mu _ {k + 1 }  ,\  \Delta
 +
^ {n} \mu _ {k}  = \Delta ^ {n - 1 } \mu _ {k} - \Delta ^ {n - 1 } \mu _ {k + 1 }  .
 +
$$
  
 
The series
 
The series
  
<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/h04674017.png" /></td> </tr></table>
+
$$
 +
\sum _ {n = 0 } ^  \infty  a _ {n}  $$
 +
 
 +
with partial sums  $  s _ {n} $
 +
is summable by the Hausdorff method to sum  $  S $
 +
if
  
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
+
$$
 +
\lim\limits _ {n \rightarrow \infty } \
 +
\sigma _ {n}  = S.
 +
$$
  
<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/h04674020.png" /></td> </tr></table>
+
The field and the regularity of the Hausdorff method depend on the sequence  $  \{ \mu _ {n} \} $.
 +
If  $  \{ \mu _ {n} \} $
 +
is a real sequence, then for the regularity of the method it is necessary and sufficient that  $  \{ \mu _ {n} \} $
 +
is the difference of two absolutely-monotone sequences and that
  
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
+
$$
 +
\lim\limits _ {m \rightarrow \infty } \
 +
\Delta  ^ {m} \mu _ {m}  = 0; \ \
 +
\mu _ {0= 1;
 +
$$
  
<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  $  \mu _ {n} $
 +
are regular moments.
  
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  $  \mu _ {n} = 1/( q + 1)  ^ {n} $
 +
the Hausdorff method turns into the Euler method  $  ( E, q) $,
 +
for  $  \mu = 1/( n + 1)  ^ {k} $
 +
into the Hölder method  $  ( H, k) $,
 +
and for
  
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
+
$$
 +
\mu _ {n}  =
 +
\frac{1}{\left ( \begin{array}{c}
 +
n + k \\
 +
k
 +
\end{array}
 +
\right ) }
  
<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/h04674030.png" /></td> </tr></table>
+
$$
  
into the Cesàro method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046740/h04674031.png" />.
+
into the Cesàro method $  ( C, k) $.
  
 
====References====
 
====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>
 
<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>

Revision as of 22:10, 5 June 2020


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

$$ t = \delta s,\ \ \tau = \mu t,\ \ \sigma = \delta t, $$

where $ \delta $ is the transformation by means of the triangular matrix $ \{ \delta _ {nk} \} $:

$$ \delta _ {nk} = \ \left \{ \begin{array}{ll} (- 1) ^ {k} \left ( \begin{array}{c} n \\ k \end{array} \right ) , & k \leq n, \\ 0 , & k > n; \\ \end{array} $$

and $ \mu $ is the diagonal transformation by means the diagonal matrix $ \| \mu _ {nk} \| $:

$$ \mu _ {nk} = \ \left \{ \begin{array}{ll} \mu _ {n} , & k = n, \\ 0, & k \neq n, \\ \end{array} \right .$$

where $ \mu _ {n} $ is a numerical sequence. The transformation

$$ \sigma = \lambda s, $$

where $ \lambda = \delta \mu \delta $, $ \{ \mu _ {n} \} $ is an arbitrary numerical sequence, is called a general Hausdorff transformation, and the matrix $ \delta \mu \delta $— a Hausdorff matrix. Written out, a general Hausdorff transformation has the form

$$ \sigma _ {n} = \ \sum _ {k = 0 } ^ { n } \lambda _ {nk} s _ {k} , $$

where

$$ \lambda _ {nk} = \ \left \{ \begin{array}{ll} \left ( \begin{array}{c} n \\ k \end{array} \right ) \Delta ^ {n - k } \mu _ {k} , & k \leq n, \\ 0, & k > n; \\ \end{array} $$

$$ \Delta \mu _ {k} = \mu _ {k} - \mu _ {k + 1 } ,\ \Delta ^ {n} \mu _ {k} = \Delta ^ {n - 1 } \mu _ {k} - \Delta ^ {n - 1 } \mu _ {k + 1 } . $$

The series

$$ \sum _ {n = 0 } ^ \infty a _ {n} $$

with partial sums $ s _ {n} $ is summable by the Hausdorff method to sum $ S $ if

$$ \lim\limits _ {n \rightarrow \infty } \ \sigma _ {n} = S. $$

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

$$ \lim\limits _ {m \rightarrow \infty } \ \Delta ^ {m} \mu _ {m} = 0; \ \ \mu _ {0} = 1; $$

or, in another terminology, necessary and sufficient is that the $ \mu _ {n} $ are regular moments.

The Hausdorff summation method contains as special cases a number of other well-known summation methods. Thus, for $ \mu _ {n} = 1/( q + 1) ^ {n} $ the Hausdorff method turns into the Euler method $ ( E, q) $, for $ \mu = 1/( n + 1) ^ {k} $ into the Hölder method $ ( H, k) $, and for

$$ \mu _ {n} = \frac{1}{\left ( \begin{array}{c} n + k \\ k \end{array} \right ) } $$

into the Cesàro method $ ( C, k) $.

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)
How to Cite This Entry:
Hausdorff summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_summation_method&oldid=47199
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article