Silverman–Toeplitz theorem

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



\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 < \infty & & \text{(The absolute row sums are bounded.)} \end{align} $$

An example is Cesaro summation, a matrix summability method with
 * $$a_{mn}=\begin{cases}\frac{1}{m} & n\le m\\ 0 & n>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},$$