Silverman–Toeplitz theorem explained

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]

(a _{i,j} ) _{i,j} with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties:\begin{align} & \lim _{i} a _{i,j} = 0 j \in N & & (Everycolumnsequenceconvergesto0.) \\[ 3 pt] & \lim _{i} \sum ^{inf ty} _{j =0} a _{i,j} = 1 & & (Therowsumsconvergeto1.) \\[ 3 pt] & \sup _{i} \sum ^{inf ty} _{j =0} \vert a _{i,j} \vert < inf ty & & (Theabsoluterowsumsarebounded.) \end{align}

An example is Cesaro summation, a matrix summability method with

a _{mn} = \begin{cases} (1)/(m) & n\le m\ 0 & n>m\end{cases} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & … \\ (1)/(2) & (1)/(2) & 0 & 0 & 0 & … \\ (1)/(3) & (1)/(3) & (1)/(3) & 0 & 0 & … \\ (1)/(4) & (1)/(4) & (1)/(4) & (1)/(4) & 0 & … \\ (1)/(5) & (1)/(5) & (1)/(5) & (1)/(5) & (1)/(5) & … \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix},

References

Further reading

• Toeplitz, Otto (1911) "Über allgemeine 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
• , 43-48.
• Book: Boos, Johann . Classical and modern methods in summability . 2000. 019850165X . New York . Oxford University Press .

Notes and References

1. https://archive.org/details/silvermantoeplit00rude Silverman–Toeplitz theorem