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 Zbl 48.2005.01 Zbl 48.2005.02
[2]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)