# 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.[1]

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

${\displaystyle \lim _{i\to \infty }a_{i,j}=0\quad j\in \mathbb {N} }$ (every column sequence converges to 0)
${\displaystyle \lim _{i\to \infty }\sum _{j=0}^{\infty }a_{i,j}=1}$ (the row sums converge to 1)
${\displaystyle \sup _{i}\sum _{j=0}^{\infty }\vert a_{i,j}\vert <\infty }$ (the absolute row sums are bounded).

## References

1. ^ Silverman-Toeplitz theorem, by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive
• 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