Hausdorff summation method

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} \} $:

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

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

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

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

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)