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 [math]\displaystyle{ (a_{i,j})_{i,j \in \mathbb{N}} }[/math] with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties:

[math]\displaystyle{ \begin{align} & \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 \lt \infty & & \text{(The absolute row sums are bounded.)} \end{align} }[/math]

An example is Cesaro summation, a matrix summability method with

[math]\displaystyle{ a_{mn}=\begin{cases}\frac{1}{m} & n\le m\\ 0 & n\gt m\end{cases} = \begin{pmatrix} 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 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}, }[/math]

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.