Silverman–Toeplitz theorem

Short description: Theorem of summability methods

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.

An infinite matrix $(a_{i,j}{)}_{i,j\in \mathbb{\mathbb{N}}}$ with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties:

$\begin{array}{rlrl}& \underset{i\to \mathrm{\infty }}{\lim }{a}_{i,j}=0j\in \mathbb{\mathbb{N}}& & \text{(Every column sequence converges to 0.)}\\ & \underset{i\to \mathrm{\infty }}{\lim }\sum _{j=0}^{\mathrm{\infty }}{a}_{i,j}=1& & \text{(The row sums converge to 1.)}\\ & \underset{i}{\sup }\sum _{j=0}^{\mathrm{\infty }}|a_{i,j}|<\mathrm{\infty }& & \text{(The absolute row sums are bounded.)}\end{array}$

An example is Cesaro summation, a matrix summability method with

$a_{mn}=\{\begin{array}{ll}\frac{1}{m}& n\le m\\ 0& n>m\end{array}=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& \cdots \\ \frac{1}{2}& \frac{1}{2}& 0& 0& 0& \cdots \\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}& 0& 0& \cdots \\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& 0& \cdots \\ \frac{1}{5}& \frac{1}{5}& \frac{1}{5}& \frac{1}{5}& \frac{1}{5}& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \\ \end{array}\right),$

References

Further reading

• 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, https://archive.org/details/divergentseries033523mbp , 43-48.
• Boos, Johann (2000). Classical and modern methods in summability. New York: Oxford University Press. ISBN 019850165X. https://books.google.com/books/about/Classical_and_Modern_Methods_in_Summabil.html?id=kZ9cy6XyidEC.