In mathematics, the Silverman–Toeplitz theorem, first proved by

Otto Toeplitz Otto Toeplitz (1 August 1881 – 15 February 1940) was a German mathematician working in functional analysis., reprinted in Life and work Toeplitz was born to a Jewish family of mathematicians. Both his father and grandfather were ''Gymnasiu ...

, is a result in

summability theory In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...

characterizing

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

summability methods that are regular. A regular matrix summability method is a matrix transformation of a

convergent sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...

which preserves the

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

.Silverman–Toeplitz theorem
by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive An

infinite matrix In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \beg ...

(a_)_

with

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

-valued entries defines a regular summability method

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...

it satisfies all of the following properties: :

& \sup_i \sum_^ \vert a_ \vert < \infty & & \text \end

An example is Cesaro summation, a matrix summability method with :

a_=\begin\frac & n\le m\\ 0 & n>m\end = \begin
1 & 0 & 0 & 0 & 0 & \cdots \\
\frac & \frac & 0 & 0 & 0 & \cdots \\
\frac & \frac & \frac & 0 & 0 & \cdots \\
\frac & \frac & \frac & \frac & 0 & \cdots \\
\frac & \frac & \frac & \frac & \frac & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\
\end,

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