Diffusion capacitance

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

$\underset{i\to \mathrm{\infty }}{\lim }{a}_{i,j}=0j\in \mathbb{\mathbb{N}}$ (every column sequence converges to 0)

$\underset{i\to \mathrm{\infty }}{\lim }\sum _{j=0}^{\mathrm{\infty }}{a}_{i,j}=1$ (the row sums converge to 1)

${\sup }_i\sum _{j=0}^{\mathrm{\infty }}\left|a_{i,j}\right|<\mathrm{\infty }$ (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