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 $(a_{i,j})_{i,j \in \mathbb{N}}$ with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties:

$\lim_{i \to \infty} a_{i,j} = 0 \quad j \in \mathbb{N}$ (every column sequence converges to 0)

$\lim_{i \to \infty} \sum_{j=0}^{\infty} a_{i,j} = 1$ (the row sums converge to 1)

$\sup_{i} \sum_{j=0}^{\infty} \vert a_{i,j} \vert < \infty$ (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