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Silverman–Toeplitz theorem : ウィキペディア英語版
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_)_ a_ = 0 \quad j \in \mathbb (every column sequence converges to 0)
:\lim_ \sum_^ a_ = 1 (the row sums converge to 1)
:\sup_ \sum_^ \vert a_ \vert < \infty (the absolute row sums are bounded).
== References ==

* Toeplitz, Otto (1911) "(''Über die 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

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