Diffusion capacitance

2013-10-06T11:31:24Z

117.199.194.157:

In [[mathematics]], the '''Silverman–Toeplitz theorem''', first proved by [[Otto Toeplitz]], is a result in [[summability theory]] characterizing [[Matrix (mathematics)|matrix]] summability methods that are regular. A regular matrix summability method is a matrix transformation of a [[convergent sequence]] which preserves the [[Limit of a sequence|limit]].

An [[infinite matrix]] <math>(a_{i,j})_{i,j \in \mathbb{N}}</math> with [[complex number|complex]]-valued entries defines a regular summability method [[if and only if]] it satisfies all of the following properties
:<math>\lim_{i \to \infty} a_{i,j} = 0 \quad j \in \mathbb{N}</math> (every column sequence converges to 0)
:<math>\lim_{i \to \infty} \sum_{j=0}^{\infty} a_{i,j} = 1</math> (the row sums converge to 1)
:<math>\sup_{i} \sum_{j=0}^{\infty} \vert a_{i,j} \vert < \infty</math> (the absolute row sums are bounded).

== References ==
* Toeplitz, Otto (1911) "[http://matwbn.icm.edu.pl/ksiazki/pmf/pmf22/pmf2219.pdf ''Über die lineare Mittelbildungen.'']" ''Prace mat.-fiz.'', '''22''', 113–118 (the original paper in [[German language|German]])
* Silverman, Louis Lazarus (1913) "On the definition of the sum of a divergent series." University of Missouri Studies, Math. Series I, 1–96

{{DEFAULTSORT:Silverman-Toeplitz theorem}}
[[Category:Theorems in analysis]]
[[Category:Summability methods]]
[[Category:Summability theory]]

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Diffusion capacitance