Silverman–Toeplitz theorem

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In mathematics the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods which are regular. A regular matrix summability method is a matrix transformation of a convergent sequence which preserves the limit.

[edit] Silverman–Toeplitz theorem

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

\lim_{i \to \infty} a_{i,j} = 0 \quad j \in \mathbb{N} (any column sequences converge to 0)
\lim_{j \to \infty} \sum_{i=0}^{\infty} a_{i,j} = 1 (the sum of the rows converges to 1)
\sup_{j} \sum_{i=0}^{\infty} \vert a_{i,j} \vert < \infty (the row sums are bounded)

[edit] References

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