Silverman–Toeplitz theorem explained

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.

(a_i,j)_i,j

with complex-valued entries defines a regular matrix summability method if and only if it satisfies all of the following properties:

\begin{align} &\lim_ia_i,j=0 j\inN&&(Everycolumnsequenceconvergesto0.)\\[3pt] &\lim_i

	infty
\sum
	j=0

a_i,j=1&&(Therowsumsconvergeto1.)\\[3pt] &\sup_i

	infty
\sum
	j=0

\verta_i,j\vert<infty&&(Theabsoluterowsumsarebounded.) \end{align}

An example is Cesàro summation, a matrix summability method with

a_mn=\begin{cases}

	1
	m

&n\lem\ 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}.

Formal statement

Let the aforementioned inifinite matrix

(a_i,j)_i,j

of complex elements satisfy the following conditions:

\lim_ia_i,j=0

for every fixed

j\inN

\sup_i

	infty
\sum
	j=0

\verta_i,j\vert<infty

;

and

z_n

be a sequence of complex numbers that converges to

\lim_nz_n=z_infty

. We denote

S_n

as the weighted sum sequence:

S_n=

	n
\sum
	m=1

{\left(a_n,z_n\right)}

Then the following results hold:

\lim_nz_n=z_infty=0

, then

\lim_n{S_n

} = 0 .

\lim_nz_n=z_infty\ne0

and

\lim_i

	infty
\sum
	j=0

a_i,j=1

, then

\lim_n{S_n

} = z_ .^[2]

Proof

Proving 1.

For the fixed

j\inN

the complex sequences

z_n

S_n

and

a_i,

approach zero if and only if the real-values sequences

\left|z_n\right|

\left|S_n\right|

and

\left|a_i,\right|

approach zero respectively. We also introduce

M=\sup_i

	infty
\sum
	j=0

\verta_i,j\vert>0

Since

\left|z_n\right|\to0

, for prematurely chosen

\varepsilon>0

there exists

N_z=N_z\left(\varepsilon\right)

, so for every

n>N_z\left(\varepsilon\right)

we have

\left|z_n\right|<

	\varepsilon
	2M

. Next, for some

N_a=N_a\left(\varepsilon\right)>N_\varepsilon\left(\varepsilon\right)

it's true, that

\left|a_n,\right|<

	M
	N_\varepsilon

for every

n>N_a\left(\varepsilon\right)

and

1\leqslantm\leqslantn

. Therefore, for every

n>N_a\left(\varepsilon\right)

\begin{align} &\left|S_n\right|=\left|

	n
\sum
	m=1

\left(a_n,z_n\right)\right|\leqslant

	n
\sum
	m=1

\left(\left|a_n,\right| ⋅ \left|z_n\right|\right) =

	N_\varepsilon
\sum
	m=1

\left(\left|a_n,\right| ⋅ \left|z_n\right|\right)+

	n
\sum
	m=N_\varepsilon

\left(\left|a_n,\right| ⋅ \left|z_n\right|\right)<\ &<N_\varepsilon ⋅

	M
	N_\varepsilon

⋅

	\varepsilon
	2M

	\varepsilon
	2M

	n
\sum
	m=N_\varepsilon

\left|a_n,\right| \leqslant

	\varepsilon
	2

	\varepsilon
	2M

	n
\sum
	m=1

\left|a_n,\right|\leqslant

	\varepsilon
	2

	\varepsilon
	2M

⋅ M=\varepsilon \end{align}

which means, that both sequences

\left|S_n\right|

and

S_n

converge zero.^[3]

Proving 2.

\lim_n\left(z_n-z_infty\right)=0

. Applying the already proven statement yields

\lim_n

	n
\sum
	m=1

(a_n,m\left(z_n-z_infty\right))=0

. Finally,

\lim_nS_n=\lim_n

	n
\sum
	m=1

(a_n,mz_n) =\lim_n

	n
\sum
	m=1

(a_n,m\left(z_n-z_infty\right))+z_infty\lim_n

	n
\sum
	m=1

(a_n,m) =0+z_infty ⋅ 1=z_infty

, which completes the proof.

References

Notes and References

https://archive.org/details/silvermantoeplit00rude Silverman–Toeplitz theorem
Linero . Antonio . Rosalsky . Andrew . 2013-07-01 . On the Toeplitz Lemma, Convergence in Probability, and Mean Convergence . live . Stochastic Analysis and Applications . en . 31 . 4 . 1 . 10.1080/07362994.2013.799406 . 0736-2994 . 2024-11-17.
Book: Ljashko, Ivan Ivanovich . Математический анализ: введение в анализ, производная, интеграл. Справочное пособие по высшей математике. . Bojarchuk . Alexey Klimetjevich . Gaj . Jakov Gavrilovich . Golovach . Grigory Petrovich . 2001 . Editorial URSS . 978-5-354-00018-0 . 1st . 1 . Moskva . 58 . ru . Mathematical analysis: the introduction into analysis, derivatives, integrals. The handbook to mathematical analysis. .