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Silverman–Toeplitz theorem

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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. 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.)}}\\&\lim _{i\to \infty }\sum _{j=0}^{\infty }a_{i,j}=1&&{\text{(The row sums converge to 1.)}}\\&\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}}.}

Formal statement

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:

  1. 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} } .
  2. sup i ∈ N ∑ j = 1 i | a i , j | < ∞ {\displaystyle \sup _{i\in \mathbb {N} }\sum _{j=1}^{i}\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:

  1. 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} .
  2. If lim n → ∞ z n = z ∞ ≠ 0 {\displaystyle \lim _{n\to \infty }z_{n}=z_{\infty }\neq 0} and lim i → ∞ ∑ j = 1 i a i , j = 1 {\displaystyle \lim _{i\to \infty }\sum _{j=1}^{i}a_{i,j}=1} , then lim n → ∞ S n = z ∞ {\displaystyle \lim _{n\to \infty }{S_{n}}=z_{\infty }} .

Proof

Proving 1.

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 = 1 i | a i , j | > 0 {\displaystyle M=\sup _{i\in \mathbb {N} }\sum _{j=1}^{i}\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.

Proving 2.

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.

References

Citations

  1. Silverman–Toeplitz theorem, by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive
  2. 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)
  3. 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.

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