Silverman–Toeplitz theorem

In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in series summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a linear sequence transformation that preserves the limits of convergent sequences.^[1] The linear sequence transformation can be applied to the divergent sequences of partial sums of divergent series to give those series generalized sums.

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

{\begin{aligned}&\lim _{i\to \infty }a_{i,j}=0\quad j\in \mathbb {N} &&{\text{(Every column sequence converges to 0.)}}\\[3pt]&\lim _{i\to \infty }\sum _{j=0}^{\infty }a_{i,j}=1&&{\text{(The row sums converge to 1.)}}\\[3pt]&\sup _{i}\sum _{j=0}^{\infty }\vert a_{i,j}\vert <\infty &&{\text{(The absolute row sums are bounded.)}}\end{aligned}}

An example is Cesàro summation, a matrix summability method with

a_{mn}={\begin{cases}{\frac {1}{m}}&n\leq m\\0&n>m\end{cases}}={\begin{pmatrix}1&0&0&0&0&\cdots \\{\frac {1}{2}}&{\frac {1}{2}}&0&0&0&\cdots \\{\frac {1}{3}}&{\frac {1}{3}}&{\frac {1}{3}}&0&0&\cdots \\{\frac {1}{4}}&{\frac {1}{4}}&{\frac {1}{4}}&{\frac {1}{4}}&0&\cdots \\{\frac {1}{5}}&{\frac {1}{5}}&{\frac {1}{5}}&{\frac {1}{5}}&{\frac {1}{5}}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \\\end{pmatrix}}.

Formal statement

Let the aforementioned inifinite matrix $(a_{i,j})_{i,j\in \mathbb {N} }$ of complex elements satisfy the following conditions:

$\lim _{i\to \infty }a_{i,j}=0$ for every fixed $j\in \mathbb {N}$ .
$\sup _{i\in \mathbb {N} }\sum _{j=1}^{i}\vert a_{i,j}\vert <\infty$ ;

and $z_{n}$ be a sequence of complex numbers that converges to $\lim _{n\to \infty }z_{n}=z_{\infty }$ . We denote $S_{n}$ as the weighted sum sequence: $S_{n}=\sum _{m=1}^{n}a_{n,m}z_{m}$ .

Then the following results hold:

If $\lim _{n\to \infty }z_{n}=z_{\infty }=0$ , then $\lim _{n\to \infty }{S_{n}}=0$ .
If $\lim _{n\to \infty }z_{n}=z_{\infty }\neq 0$ and $\lim _{i\to \infty }\sum _{j=1}^{i}a_{i,j}=1$ , then $\lim _{n\to \infty }{S_{n}}=z_{\infty }$ .^[2]

Proof

Proving 1.

For the fixed $j\in \mathbb {N}$ the complex sequences $z_{n}$ , $S_{n}$ and $a_{i,j}$ approach zero if and only if the real-values sequences $\left|z_{n}\right|$ , $\left|S_{n}\right|$ and $\left|a_{i,j}\right|$ approach zero respectively. We also introduce $M=1+\sup _{i\in \mathbb {N} }\sum _{j=1}^{i}\vert a_{i,j}\vert >0$ .

Since $\left|z_{n}\right|\to 0$ , for prematurely chosen $\varepsilon >0$ there exists $N_{\varepsilon }\in \mathbb {N}$ , so for every $n>N_{\varepsilon }$ we have $\left|z_{n}\right|<{\frac {\varepsilon }{2M}}$ . Next, for some $N_{a}=N_{a}\left(\varepsilon \right)>N_{\varepsilon }$ it's true, that $\sum _{m=1}^{n}|a_{n,m}|<{\frac {\varepsilon }{2\left(\max _{m\leq N_{\varepsilon }}|z_{m}|+1\right)}}$ for every $n>N_{a}\left(\varepsilon \right)$ . Therefore, for every $n>N_{a}\left(\varepsilon \right)$

${\begin{aligned}&\left|S_{n}\right|=\left|\sum _{m=1}^{n}\left(a_{n,m}z_{m}\right)\right|\leqslant \sum _{m=1}^{n}\left(\left|a_{n,m}\right|\cdot \left|z_{m}\right|\right)=\sum _{m=1}^{N_{\varepsilon }}\left(\left|a_{n,m}\right|\cdot \left|z_{m}\right|\right)+\sum _{m=N_{\varepsilon }+1}^{n}\left(\left|a_{n,m}\right|\cdot \left|z_{m}\right|\right)<\\&<\max _{1\leq m\leq N_{\varepsilon }}(|z_{m}|)\cdot \sum _{m=1}^{N_{\varepsilon }}|a_{n,m}|+{\frac {\varepsilon }{2M}}\sum _{m=N_{\varepsilon }+1}^{n}\left|a_{n,m}\right|\leqslant {\frac {\varepsilon }{2}}+{\frac {\varepsilon }{2M}}\sum _{m=1}^{n}\left|a_{n,m}\right|\leqslant {\frac {\varepsilon }{2}}+{\frac {\varepsilon }{2M}}\cdot M=\varepsilon \end{aligned}}$

which means, that both sequences $\left|S_{n}\right|$ and $S_{n}$ converge zero.^[3]

Proving 2.

$\lim _{n\to \infty }\left(z_{m}-z_{\infty }\right)=0$ . Applying the already proven statement yields . Finally, $\lim _{n\to \infty }\sum _{m=1}^{n}{\big (}a_{n,m}\left(z_{m}-z_{\infty }\right){\big )}=0$

$\lim _{n\to \infty }S_{n}=\lim _{n\to \infty }\sum _{m=1}^{n}{\big (}a_{n,m}z_{m}{\big )}=\lim _{n\to \infty }\sum _{m=1}^{n}{\big (}a_{n,m}\left(z_{m}-z_{\infty }\right){\big )}+z_{\infty }\lim _{n\to \infty }\sum _{m=1}^{n}{\big (}a_{n,m}{\big )}=0+z_{\infty }\cdot 1=z_{\infty }$ , which completes the proof.

References

Citations

↑ Silverman–Toeplitz theorem, by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive
↑ Linero, Antonio; Rosalsky, Andrew (2013-07-01). "On the Toeplitz Lemma, Convergence in Probability, and Mean Convergence" (PDF). Stochastic Analysis and Applications. 31 (4): 684–694. doi:10.1080/07362994.2013.799406. ISSN 0736-2994. Retrieved 2024-11-17.
↑ Ljashko, Ivan Ivanovich; Bojarchuk, Alexey Klimetjevich; Gaj, Jakov Gavrilovich; Golovach, Grigory Petrovich (2001). Математический анализ: введение в анализ, производная, интеграл. Справочное пособие по высшей математике [Mathematical analysis: the introduction into analysis, derivatives, integrals. The handbook to mathematical analysis.] (in Russian). Vol. 1 (1st ed.). Moskva: Editorial URSS. p. 58. ISBN 978-5-354-00018-0.