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Questions tagged [summability-theory]

For questions about summation/resummation/regularization methods, ways to assign meaningful values to divergent sum. Tauberian theorems for summation methods and other theoretical and applicational results.

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Is there a way to read a summability property of a family of functions from the behavior of coefficients of the corresponding Fourier transform?

If $(K_N)_{N\in\mathbb{N}}$ (or indexed by any other directed set) is a family of functions in $L^1(\mathbb{T})$ defined on the 1-torus $\mathbb{T}$ such that: $$\forall p\in[1,+\infty), \forall f\in ...
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Stability of matrix summability methods

I'm currently studying summabiltiy method defined by infinite matrices and I'd like to find a characterization of stable methods. Suppose $A$ is an infinite matrix, $x$ a real (or complex) sequence, ...
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Boundedness of a sequence in Orlicz Space

By the Definition of Orlicz function $M$, we know that $M(0)=0$, $M(x)>0$ for $x>0$ and $M(x)\rightarrow \infty$ as $x\rightarrow \infty$. And the Orlicz-Luxemborg norm is given by $\|x\|=Inf\{r&...
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72 views

Euler Sequence Space

In the Euler sequence $\sum_{k=0}^{n}\binom{n}{k}(1-\alpha)^{n-k}\alpha^k=1$ by binomial law. However, I encountered another sum $$\sum_{n=k}^{\infty}\binom{n}{k}(1-\alpha)^{n-k}\alpha^k=\frac{1}{\...
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80 views

Do the partial sums of a divergent series converge to Cesaro or Abel sums in some metric?

Let $(a_n)$ be a sequence in $\mathbb{R}$, and let $s_n$ be the $n^{th}$ partial sum of the sequence. Then the Cesaro sum of $(a_n)$ is the limit of the average of the first $n$ partial sums as $n$ ...
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Show that if $\sum_{n=0}^{\infty}a_n^k=\sum_{n=0}^{\infty}b_n^k$ for all $k\in \Bbb N^*$ then $(a_n) = (b_n)$

Let $(a_n),(b_n)$ two summable sequences. Show that if $\sum_{n=0}^{\infty}a_n^k=\sum_{n=0}^{\infty}b_n^k$ for all $k\in \Bbb N^*$ then $(a_n) = (b_n)$ more or less a permutation What I tried: to ...
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1answer
58 views

About extracting a summable subsequence from a nonsummable sequence

Let $(a_k)_{k \in \mathbb{N}} \subset ]0,+\infty[$, and assume that $a_k \to 0$ as $k$ goes to $+\infty$, but $a_k \notin \ell^1(\mathbb{N})$. It is easy to prove that we can extract a subsequence $(...
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2answers
76 views

Sum of a geometric series over natural density zero indices

It is well known that the geometric power series $% %TCIMACRO{\dsum \limits_{k=0}^{\infty}}% %BeginExpansion {\displaystyle \sum \limits_{k=0}^{\infty}} %EndExpansion x^{k}$ is convergent to $\dfrac{1}...
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An oscillating series of real numbers which can be resummed to a complex value

Okay so here I go again studying summability theory I was wondering the following problem but first I'll state a few conventions: A series diverges if the partial sums tends to $\pm \infty$, ...
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A Summability methods which sum the harmonic series

Studying summability theory I've come across many summation methods however by now I know only two not very interesting method which re-sums the harmonic series $\sum_{n=0}^\infty \frac 1{n+1}$ : the ...
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172 views

Proof Silverman-Toeplitz theorem

Proof Silverman-Toeplitz theorem: Let $A$ be an infinite matrix with entries $(a_{ij})$. Two sequences $\sigma $ and $s$ are related by this matrix as follows $$\sigma_i =\sum_{j=0}^{\infty} a_{ij}s_i$...
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If $E$ is a normed $ℝ$-vector space, are we able to show that $(x_i)_{i∈I}⊆E$ is summable $⇔$ $\left(\left\|x_i\right\|_E\right)_{i∈I}$ is summable?

Let $I\ne\emptyset$ be a set $E$ be a normed $\mathbb R$-vector space $(x_i)_{i\in I}\subseteq E$ is called summable with sum $x\in E$ $:\Leftrightarrow$ $$\forall\varepsilon>0:\exists J\subseteq ...
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1answer
69 views

Is the series corresponding to a summable family absolutely convergent?

Let $I\ne\emptyset$ be a set $E$ be a normed $\mathbb R$-vector space $(x_i)_{i\in I}\subseteq E$ is called summable with sum $x\in E$ $:\Leftrightarrow$ $$\forall\varepsilon>0:\exists J\subseteq ...
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1answer
220 views

Every linear and regular summation method is also stable

In the theory of divergent series regularization I've came to the following conclusion and I would like to know if my considerations are right or not. First I'll recall some definitions: A summation ...
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60 views

Triangle inequality and summation methods for divergent series

I'm focusing on summability theory/summation methods/resummation theory and related topics, in other words methods to give meaningful sum to divergent series. First I recall that a method is called ...
2
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1answer
296 views

Silverman-Toeplitz Theorem

Reading this paper On Deferred Statistical Convergence of Sequences by Kucukaslan and Yilmazturk published in KYUNGPOOK Math. J.56(2016), 357-366 I am stuck at theorem $2.2.7$ I get $\{n^{(1)} ,n^{(2)...
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1answer
356 views

Where to find a proof of Silverman-Toeplitz?

I am referring to the theorem which gives a necessary and sufficient condition on a infinite matrix that maps convergent sequences to sequences converging to the same limits. Wiki gives a link to ...