Regular summation methods

permanent summation methods

2020 Mathematics Subject Classification: Primary: 40C [MSN][ZBL]

Regular summation methods are those methods for summing series (sequences) that sum every convergent series (sequence) to the same sum as that to which it converges. Regular summation methods are a special case of conservative summation methods, which sum every convergent series (sequence) to a finite sum, although possibly different from that to which it converges. If a regular summation method is defined by the transformation of a sequence $(s_n)$ to a sequence $(\sigma_n)$ by means of an infinite matrix $[a_{nk}]$: \begin{equation} \label{eq1} \sigma_n = \sum_{k=1}^{\infty} a_{nk}s_k, \quad n=1,2,\ldots \end{equation} (see Matrix summation method), then the transformation \ref{eq1} and the matrix of this transformation, $[a_{nk}]$, are called regular.

Many of the most common summation methods are regular. This applies to the Cesàro summation methods $(C,k)$ for $k \geq 0$, the Hölder summation methods and the Abel summation method, among others. There are non-regular summation methods, such as the Cesàro summation method $(C,k)$ for $k<0$, and the Riemann summation method.

A summation method is called completely regular if it is regular and if it sums every series (sequence) with real terms converging to $\infty$ (or $-\infty$) to $\infty$ (respectively, $-\infty$). A regular summation method defined by a positive matrix is completely regular (see also Regularity criteria).

References

[Ba]	S. Baron, "Introduction to theory of summation of series", Tallin (1977) (In Russian)
[Co]	R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950)
[Ha]	G.H. Hardy, "Divergent series", Clarendon Press (1949)
[Ka]	G.F. Kangro, "Theory of summability of sequences and series" J. Soviet Math., 5 : 1 (1976) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal., 12 (1974) pp. 5–70