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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Are the sequences of families for summability or improper summability:
I would like to analyse the following sequences (a, b and c) of families for summability or improper summability:
a) $ a_{i}:=i^{-\alpha} $ for $ \alpha>0 $ :
b) $a_{i}:=(-1)^{i} i^{-\alpha}$ for $\...
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Analytic continuation of a lambert series $f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$?
Let's define a function for complex $z$ :
$$f(z)=\sum_{n=1}^\infty \frac{2^n z^n}{1-(z/2)^n}$$
Lambert series often have an analytic continuation and can even be entire functions so the poles can be ...
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A question on summable families in topological groups
I want to prove the following result involving summable families in abelian $T_2$ topological groups.
Let $u:G\to G'$ be a continuous group homomorphism between two abelian $T_2$ topological group, ...
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Product of two locally integrable function is locally integrable?
I have to solve this question :
Let f and g be two locally summable functions (f and g $\in L^1_{loc}$), show that $\int_0^x | f(t)g(x-t) |dt < +\infty$
I was thinking of showing that the function ...
- 35
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Generalizing Borel Summation to Matrices
I’m very new to this forum, I apologize in advance if I make any mistakes of not abiding by the forum rules.
I am trying to prove the regularity of Borel summation method when we have a series of ...
- 21
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If $\mu$ is a finite measure on a metric space, then for all $\varepsilon$, there is a $\delta<\delta$ with $\mu(\partial B_\delta(x))=0$
The following should be rather elementary, but since I never found a result of this kind in the literature and I never thought about this before, I wonder whether I'm missing something or not.
First ...
- 13.2k
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Integrability with respect to counting measure
I would like to prove the following fact:
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a positive function and $\mu$ the counting measure $\mu(a)=\begin{cases} \vert A \vert & \mbox{if } A\mbox{ is ...
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Representation of a bounded linear operator $T: c \to c$.
Let $T: c\to c$ be a bounded linear operator, where $c$ is the vector space of convergent real sequences. How can we prove that there exists an infinite matrix $A=(a_{n,k}: n,k\ge 1)$ such that $T(x)=...
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Can we rearrange the alternating harmonic series and make it equal to arbitrary real number?
By "Riemann Rearrangement Theorem", we can rearrange the conditionally convergence series and make it equal to arbitrary value.
And Riemann showed that rearranging the alternating harmonic series by $...
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An absolutely summable series but not 1-summable? [closed]
I am trying to find an example of an absolutely summable series ($\Sigma|{a_j}|<\infty$) but not 1-summable series ($\Sigma j|{a_j}|<\infty$).
I know that all 1-summable series are absolutely-...
- 123
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Is square of Poisson kernel conditionally convergent?
Poisson kernel is defined for $\theta\in [0,2\pi)$ as
$$P_r(\theta) = \sum\limits_{n\in\mathbb{Z}} r^{|n|}e^{in\theta}$$
which equals
$$P_r(\theta) = \frac{1-r^2}{r^2-2r\cos (\theta) + 1 }.$$
For this ...
- 750
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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:
$$\forall p\in[1,+\infty), \forall f\in ...
- 5,603
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Boundedness of a sequence in Orlicz Space
By the Definition of Orlicz function $M$, we know that $M(0)=0$, $M(x)>0$ for $x>0$ and $M(x)\rightarrow \infty$ as $x\rightarrow \infty$. And the Orlicz-Luxemborg norm is given by $$\|x\|=\inf\...
- 113
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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}{\...
- 113
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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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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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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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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 $x\in E$ $:\Leftrightarrow$ $$\forall\varepsilon>0:\exists J\subseteq ...
- 13.2k
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Is the series corresponding to a summable family absolutely convergent?
Let
$I\ne\emptyset$ be a set
$E$ be a normed $\mathbb R$-vector space
$(x_i)_{i\in I}\subseteq E$ is called summable with sum $x\in E$ $:\Leftrightarrow$ $$\forall\varepsilon>0:\exists J\subseteq ...
- 13.2k
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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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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 ...
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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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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 ...
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