In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a matrix transformation of a convergent sequence which preserves the limit.[1]
An infinite matrix with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties:
![{\displaystyle {\begin{aligned}&\lim _{i\to \infty }a_{i,j}=0\quad j\in \mathbb {N} &&{\text{(Every column sequence converges to 0.)}}\\[3pt]&\lim _{i\to \infty }\sum _{j=0}^{\infty }a_{i,j}=1&&{\text{(The row sums converge to 1.)}}\\[3pt]&\sup _{i}\sum _{j=0}^{\infty }\vert a_{i,j}\vert <\infty &&{\text{(The absolute row sums are bounded.)}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e27963b6a547742832d39a7e1f52ae3a02181586)
An example is Cesaro summation, a matrix summability method with
References
[edit]Citations
[edit]- ^ Silverman–Toeplitz theorem, by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive
Further reading
[edit]- Toeplitz, Otto (1911) "Über allgemeine lineare Mittelbildungen." Prace mat.-fiz., 22, 113–118 (the original paper in German)
- Silverman, Louis Lazarus (1913) "On the definition of the sum of a divergent series." University of Missouri Studies, Math. Series I, 1–96
- Hardy, G. H. (1949), Divergent Series, Oxford: Clarendon Press, 43-48.
- Boos, Johann (2000). Classical and modern methods in summability. New York: Oxford University Press. ISBN 019850165X.