Absolute summability

A special type of summability of series and sequences, which differs from ordinary summability in that additional restrictions are imposed. For matrix summation methods the requirement is that the series and sequences obtained as a result of the transformation corresponding to the given summation method must be absolutely convergent (cf. Matrix summation method). Let a summation method $A$ be defined as a transformation of sequences $\{s_n\}$ into sequences $\{\sigma_n\}$ by means of the matrix $\|a_{nk}\|$:

$$\sigma_n=\sum_{k=0}^\infty a_{nk}s_k,\quad n=0,1,\mathinner{\ldotp\ldotp\ldotp\ldotp}$$

The sequence $\{s_n\}$ is then absolutely summable by the method $A$ ($A$-summable) to the limit $s$ if it is $A$-summable to this limit, i.e. if

$$\lim_{n\to\infty}\sigma_n=s,$$

and if the sequence $\{\sigma_n\}$ is of bounded variation:

$$\sum_{n=1}^\infty|\sigma_n-\sigma_{n-1}|<\infty.\label{1}\tag{1}$$

If the $s_n$ are the partial sums of a series

$$\sum_{k=0}^\infty u_k,\label{2}\tag{2}$$

then the series \eqref{2} is absolutely summable by the method $A$ to the sum $s$. Condition \eqref{1} is an additional requirement which makes absolute summability different from ordinary summability. Absolute summability is defined in a similar manner for methods involving matrix transformations of series into sequences. If the summation method is defined by a transformation of the series \eqref{2} into a series

$$\sum_{n=0}^\infty\alpha_n\label{3}\tag{3}$$

by means of a matrix $\|b_{nk}\|$:

$$\alpha_n=\sum_{k=0}^\infty b_{nk}u_k,$$

then the supplementary restriction is absolute convergence of the series \eqref{3}. In the particular case in which the method $A$ corresponds to the identity transformation of a sequence into a sequence (or of a series into a series), absolute summability of a series is the same as absolute convergence.

The supplementary requirements are suitably modified for non-matrix summation methods. Thus, in the case of the Abel summation method such a requirement is that the function

$$f(x)=\sum_{k=0}^\infty u_kx^k$$

is of bounded variation on the semi-interval $0\leq x<1$. In the case of integral summation methods, absolute summability is distinguished by the requirement of absolute convergence of the corresponding integrals. Thus, for the Borel summation method the integral

$$\int\limits_0^\infty e^{-x}\sum_{k=0}^\infty\frac{u_kx^k}{k!}dx$$

must be absolutely convergent.

A summation method is said to preserve absolute convergence of a series if it absolutely sums each absolutely convergent series. If each such series is summable by this method to a sum equal to the sum to which it converges, then the method is called absolutely regular. For instance, the Cesàro summation methods $(C,k)$ are absolutely regular for $k\geq0$. The Abel method is absolutely regular. The following are necessary and sufficient conditions for the absolute regularity of a summation method defined by a transformation of a series into a series by means of a matrix $\|b_{nk}\|$:

$$\sum_{n=0}^\infty|b_{nk}|\leq M,\quad\sum_{n=0}^\infty b_{nk}=1,\quad k=0,1,\dots$$

(the Knopp–Lorentz theorem). Summation methods by means of other types of transformations are subject to analogous restrictions.

A generalization of absolute summability is absolute summability of degree $p$ where $p\geq1$. Here the condition

$$\sum_{n=1}^\infty n^{p-1}|\sigma_n-\sigma_{n-1}|^p<\infty.$$

is the supplementary restriction by which absolute summability of degree $p$ is distinguished from ordinary summability in the case of, say, a summation method defined by a transformation of a sequence $\{s_n\}$ into a sequence $\{\sigma_n\}$.

The concept of absolute summability was introduced by E. Borel for one of his methods, but was stated differently from its modern formulation: absolute summability was distinguished by imposing the condition

$$\int\limits_0^\infty e^{-x}\left|\sum_{k=0}^\infty\frac{u_{k+p}x^k}{k!}\right|dx<\infty$$

for each $p=0,1,\dots$. Absolute summability was initially used in the study of the summability of power series outside the disc of convergence. In view of the problems involved in the multiplication of summable series, absolute summability was defined and studied by the method of Cesàro ($(|C|,|k|)$-summability). The general definition of absolute summability is more recent, and has found extensive application in studies on the summation of Fourier series.

References

[Ha]	G.H. Hardy, "Divergent series", Clarendon Press (1949) MR0030620 Zbl 0032.05801
[Ka]	G.F. Kangro, "Summability theory of series and sequences" Itogi Nauk. i Tekhn. Mat. Anal., 12 (1974) pp. 5–70 (In Russian) Zbl 0339.40007
[KnLo]	K. Knopp, G.G. Lorentz, "Beiträge zur absoluten Limitierung" Arch. Math. (Basel), 2 (1949–1950) pp. 10–16
[Ko]	E. Kogbetliantz, "Sommation des séries et intégrales divergentes par les moyennes arithmétiques et typiques", Gauthier-Villars (1931) Zbl 0003.00701 JFM Zbl 57.1376.02
[ZeBe]	K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren", Springer (1970) MR0264267 Zbl 0199.11301