Theorem of summability methods
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
∈
N
{\displaystyle (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:
lim
i
→
∞
a
i
,
j
=
0
j
∈
N
(Every column sequence converges to 0.)
lim
i
→
∞
∑
j
=
0
∞
a
i
,
j
=
1
(The row sums converge to 1.)
sup
i
∑
j
=
0
∞
|
a
i
,
j
|
<
∞
(The absolute row sums are bounded.)
{\displaystyle {\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
m
n
=
{
1
m
n
≤
m
0
n
>
m
=
(
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
⋯
⋮
⋮
⋮
⋮
⋮
⋱
)
.
{\displaystyle 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}}.}
Let the aforementioned inifinite matrix
(
a
i
,
j
)
i
,
j
∈
N
{\displaystyle (a_{i,j})_{i,j\in \mathbb {N} }}
of complex elements satisfy the following conditions:
lim
i
→
∞
a
i
,
j
=
0
{\displaystyle \lim _{i\to \infty }a_{i,j}=0}
for every fixed
j
∈
N
{\displaystyle j\in \mathbb {N} }
.
sup
i
∈
N
∑
j
=
0
∞
|
a
i
,
j
|
<
∞
{\displaystyle \sup _{i\in \mathbb {N} }\sum _{j=0}^{\infty }\vert a_{i,j}\vert <\infty }
;
and
z
n
{\displaystyle z_{n}}
be a sequence of complex numbers that converges to
lim
n
→
∞
z
n
=
z
∞
{\displaystyle \lim _{n\to \infty }z_{n}=z_{\infty }}
. We denote
S
n
{\displaystyle S_{n}}
as the weighted sum sequence:
S
n
=
∑
m
=
1
n
(
a
n
,
m
z
n
)
{\displaystyle S_{n}=\sum _{m=1}^{n}{\left(a_{n,m}z_{n}\right)}}
.
Then the following results hold:
If
lim
n
→
∞
z
n
=
z
∞
=
0
{\displaystyle \lim _{n\to \infty }z_{n}=z_{\infty }=0}
, then
lim
n
→
∞
S
n
=
0
{\displaystyle \lim _{n\to \infty }{S_{n}}=0}
.
If
lim
n
→
∞
z
n
=
z
∞
≠
0
{\displaystyle \lim _{n\to \infty }z_{n}=z_{\infty }\neq 0}
and
lim
i
→
∞
∑
j
=
0
∞
a
i
,
j
=
1
{\displaystyle \lim _{i\to \infty }\sum _{j=0}^{\infty }a_{i,j}=1}
, then
lim
n
→
∞
S
n
=
z
∞
{\displaystyle \lim _{n\to \infty }{S_{n}}=z_{\infty }}
.[ 2]
For the fixed
j
∈
N
{\displaystyle j\in \mathbb {N} }
the complex sequences
z
n
{\displaystyle z_{n}}
,
S
n
{\displaystyle S_{n}}
and
a
i
,
j
{\displaystyle a_{i,j}}
approach zero if and only if the real-values sequences
|
z
n
|
{\displaystyle \left|z_{n}\right|}
,
|
S
n
|
{\displaystyle \left|S_{n}\right|}
and
|
a
i
,
j
|
{\displaystyle \left|a_{i,j}\right|}
approach zero respectively. We also introduce
M
=
sup
i
∈
N
∑
j
=
0
∞
|
a
i
,
j
|
>
0
{\displaystyle M=\sup _{i\in \mathbb {N} }\sum _{j=0}^{\infty }\vert a_{i,j}\vert >0}
.
Since
|
z
n
|
→
0
{\displaystyle \left|z_{n}\right|\to 0}
, for prematurely chosen
ε
>
0
{\displaystyle \varepsilon >0}
there exists
N
ε
=
N
ε
(
ε
)
{\displaystyle N_{\varepsilon }=N_{\varepsilon }\left(\varepsilon \right)}
, so for every
n
>
N
ε
(
ε
)
{\displaystyle n>N_{\varepsilon }\left(\varepsilon \right)}
we have
|
z
n
|
<
ε
2
M
{\displaystyle \left|z_{n}\right|<{\frac {\varepsilon }{2M}}}
. Next, for some
N
a
=
N
a
(
ε
)
>
N
ε
(
ε
)
{\displaystyle N_{a}=N_{a}\left(\varepsilon \right)>N_{\varepsilon }\left(\varepsilon \right)}
it's true, that
|
a
n
,
m
|
<
M
N
ε
{\displaystyle \left|a_{n,m}\right|<{\frac {M}{N_{\varepsilon }}}}
for every
n
>
N
a
(
ε
)
{\displaystyle n>N_{a}\left(\varepsilon \right)}
and
1
⩽
m
⩽
n
{\displaystyle 1\leqslant m\leqslant n}
. Therefore, for every
n
>
N
a
(
ε
)
{\displaystyle n>N_{a}\left(\varepsilon \right)}
|
S
n
|
=
|
∑
m
=
1
n
(
a
n
,
m
z
n
)
|
⩽
∑
m
=
1
n
(
|
a
n
,
m
|
⋅
|
z
n
|
)
=
∑
m
=
1
N
ε
(
|
a
n
,
m
|
⋅
|
z
n
|
)
+
∑
m
=
N
ε
n
(
|
a
n
,
m
|
⋅
|
z
n
|
)
<
<
N
ε
⋅
M
N
ε
⋅
ε
2
M
+
ε
2
M
∑
m
=
N
ε
n
|
a
n
,
m
|
⩽
ε
2
+
ε
2
M
∑
m
=
1
n
|
a
n
,
m
|
⩽
ε
2
+
ε
2
M
⋅
M
=
ε
{\displaystyle {\begin{aligned}&\left|S_{n}\right|=\left|\sum _{m=1}^{n}\left(a_{n,m}z_{n}\right)\right|\leqslant \sum _{m=1}^{n}\left(\left|a_{n,m}\right|\cdot \left|z_{n}\right|\right)=\sum _{m=1}^{N_{\varepsilon }}\left(\left|a_{n,m}\right|\cdot \left|z_{n}\right|\right)+\sum _{m=N_{\varepsilon }}^{n}\left(\left|a_{n,m}\right|\cdot \left|z_{n}\right|\right)<\\&<N_{\varepsilon }\cdot {\frac {M}{N_{\varepsilon }}}\cdot {\frac {\varepsilon }{2M}}+{\frac {\varepsilon }{2M}}\sum _{m=N_{\varepsilon }}^{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
|
S
n
|
{\displaystyle \left|S_{n}\right|}
and
S
n
{\displaystyle S_{n}}
converge zero.[ 3]
lim
n
→
∞
(
z
n
−
z
∞
)
=
0
{\displaystyle \lim _{n\to \infty }\left(z_{n}-z_{\infty }\right)=0}
. Applying the already proven statement yields
lim
n
→
∞
∑
m
=
1
n
(
a
n
,
m
(
z
n
−
z
∞
)
)
=
0
{\displaystyle \lim _{n\to \infty }\sum _{m=1}^{n}{\big (}a_{n,m}\left(z_{n}-z_{\infty }\right){\big )}=0}
. Finally,
lim
n
→
∞
S
n
=
lim
n
→
∞
∑
m
=
1
n
(
a
n
,
m
z
n
)
=
lim
n
→
∞
∑
m
=
1
n
(
a
n
,
m
(
z
n
−
z
∞
)
)
+
z
∞
lim
n
→
∞
∑
m
=
1
n
(
a
n
,
m
)
=
0
+
z
∞
⋅
1
=
z
∞
{\displaystyle \lim _{n\to \infty }S_{n}=\lim _{n\to \infty }\sum _{m=1}^{n}{\big (}a_{n,m}z_{n}{\big )}=\lim _{n\to \infty }\sum _{m=1}^{n}{\big (}a_{n,m}\left(z_{n}-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.
^ 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): 1. doi :10.1080/07362994.2013.799406 . ISSN 0736-2994 . Retrieved 2024-11-17 . {{cite journal }}
: CS1 maint: url-status (link )
^ 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 .
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
Hardy, G. H. (1949), Divergent Series , Oxford: Clarendon Press , 43-48.
Boos, Johann (2000). Classical and modern methods in summability . New York: Oxford University Press. ISBN 019850165X .