Silverman–Toeplitz theorem

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Theorem of summability methods

title: "Silverman–Toeplitz theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["theorems-in-mathematical-analysis", "summability-methods", "summability-theory"] description: "Theorem of summability methods" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Silverman–Toeplitz_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Theorem of summability methods ::

In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in series summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a linear sequence transformation that preserves the limits of convergent sequences. The linear sequence transformation can be applied to the divergent sequences of partial sums of divergent series to give those series generalized sums.

An infinite matrix (a_{i,j})_{i,j \in \mathbb{N}} with complex-valued entries defines a regular matrix 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 \end{align}

An example is Cesàro summation, a matrix summability method with :a_{mn}=\begin{cases}\frac{1}{m} & n\le m\ 0 & nm\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}.

Formal statement

Let the aforementioned inifinite matrix (a_{i,j})_{i,j \in \mathbb{N}} of complex elements satisfy the following conditions:

\lim_{i \to \infty} a_{i,j} = 0 for every fixed j \in \mathbb{N} .

  1. \sup_{i \in \mathbb{N}} \sum_{j=1}^{i} \vert a_{i,j} \vert ;

and z_{n} be a sequence of complex numbers that converges to \lim_{n \to \infty} z_{n} = z_{\infty} . We denote S_{n} as the weighted sum sequence: S_{n} = \sum_{m = 1}^{n} a_{n, m} z_{m} .

Then the following results hold:

  1. If \lim_{n \to \infty} z_{n} = z_{\infty} = 0 , then \lim_{n \to \infty} {S_{n}} = 0 .
  2. If \lim_{n \to \infty} z_{n} = z_{\infty} \ne 0 and \lim_{i \to \infty} \sum_{j=1}^{i} a_{i,j} = 1 , then \lim_{n \to \infty} {S_{n}} = z_{\infty} .

Proof

Proving 1.

For the fixed j \in \mathbb{N} the complex sequences z_{n} , S_{n} and a_{i, j} approach zero if and only if the real-values sequences \left| z_{n} \right| , \left| S_{n} \right| and \left| a_{i, j} \right| approach zero respectively. We also introduce M = 1 + \sup_{i \in \mathbb{N}} \sum_{j=1}^{i} \vert a_{i,j} \vert 0.

Since \left| z_{n} \right| \to 0 , for prematurely chosen \varepsilon 0 there exists N_{\varepsilon} \in \mathbb{N}, so for every n N_{\varepsilon} we have \left| z_{n} \right| . Next, for some N_{a} = N_{a}\left( \varepsilon \right ) N_{\varepsilon} it's true, that \sum_{m=1}^n |a_{n, m} | for every n N_{a}\left( \varepsilon \right ) . Therefore, for every n N_{a}\left( \varepsilon \right )

\begin{align} & \left| S_{n} \right| = \left| \sum_{m = 1}^{n} \left( a_{n, m} z_{m} \right) \right| \leqslant \sum_{m = 1}^{n} \left( \left| a_{n, m} \right| \cdot \left| z_{m} \right| \right) = \sum_{m = 1}^{N_{\varepsilon}} \left( \left| a_{n, m} \right| \cdot \left| z_{m} \right| \right) + \sum_{m = N_{\varepsilon}+1}^{n} \left( \left| a_{n, m} \right| \cdot \left| z_{m} \right| \right) & \leqslant \frac {\varepsilon} {2} + \frac {\varepsilon} {2M} \sum_{m = 1}^{n} \left| a_{n, m} \right| \leqslant \frac {\varepsilon} {2} + \frac {\varepsilon} {2M} \cdot M = \varepsilon \end{align}

which means, that both sequences \left| S_{n} \right| and S_{n} converge zero.

Proving 2.

\lim_{n \to \infty} \left( z_{m} - z_{\infty} \right) = 0 . Applying the already proven statement yields . Finally,\lim_{n \to \infty} \sum_{m=1}^{n} \big( a_{n,m} \left( z_{m} - z_{\infty} \right) \big) = 0

\lim_{n \to \infty} S_{n} = \lim_{n \to \infty} \sum_{m=1}^{n} \big( a_{n,m} z_{m} \big) = \lim_{n \to \infty} \sum_{m=1}^{n} \big( a_{n,m} \left( z_{m} - z_{\infty} \right) \big) + z_{\infty} \lim_{n \to \infty} \sum_{m=1}^{n} \big( a_{n,m} \big) = 0 + z_{\infty} \cdot 1 = z_{\infty} , which completes the proof.

References

Citations

References

  1. [https://archive.org/details/silvermantoeplit00rude Silverman–Toeplitz theorem], by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive
  2. (2013-07-01). "On the Toeplitz Lemma, Convergence in Probability, and Mean Convergence". Stochastic Analysis and Applications.
  3. (2001). "Математический анализ: введение в анализ, производная, интеграл. Справочное пособие по высшей математике.". Editorial URSS.

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