Questions tagged [cesaro-summable]
For questions about Cesàro summation and Cesàro summable sequences.
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Is Cesàro limit of a stochastic matrix always Block diagonalized with Block being either diagonalized or rank 1 (up to some permutation)?
I am interesting in understanding the possible Cesàro limit of a stochastic matrix A.
It is known that the Cesàro limit $P=\lim_{n\rightarrow \infty}P_n=\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^...
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Does the rate $\log n$ imply "almost harmonic"?
Let $\{c_k\}$ be a decreasing positive sequence such that $\sum_{k=1}^n c_k \sim \log n$. Does it say $c_k=O(1/k)?$
I have found a reference where it says that the converse is true. I tried to tackle ...
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Differently defined Cesàro summability implies Abel summability
I am trying to solve the Exercise 10 of Section 5.2 of the book `Multiplicative Number Theory I. Classical Theory' by Montgomery & Vaughan. In the exercise, they define the Cesàro summability of ...
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Derivation of formula for (C,1) summability of integrals
I'm having trouble understanding why the (C,1) formula for integrals is given by $$ \lim_{\lambda\to\infty} \int_{0}^{\lambda}\bigg(1- \frac{x}{\lambda} \bigg)f(x) \,dx $$
I understand that we want ...
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Proof that the sequence of cesaro means converges to the same limit as the sequence.
The question is: Show that if $(x_n)$ is a convergent sequence, then the sequence given by the averages $y_{n}=\frac{(x_{1}+x_{2}+...+x_{n})}{n}$ also converges to the same limit.
We know $|x_{n}-x_{0}...
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Convergence of Cesaro means
Let $T$ be a linear operator on a dual Banach space $X'$ such that $M:=\sup_{n\in\mathbb N}\|T^n\| <\infty$. Define $$A_nx = \frac1n \sum_{j=0}^{n-1} T^jx\qquad (n\in\mathbb N, x\in X).$$
Show that ...
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Cesaro $(C,\alpha)$ summable implies Abel summable.
I've found quite a few questions regarding the statement "Cesaro $(C,1)$ summability implies Abel summability", e.g. this question, but haven't been able to find a proof for higher Cesaro ...
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Sequence converges, then the mean also converges (withouht knowing the mean converges to the same value)
Maybe this question has been read a lot of timehere but I failed to find this kind of version. Let me explain.
Let $\{a_n\}$ be a convergent sequence, then $\left \{ \frac{S_n}{n} \right\}$ also ...
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Confusion on Baby Rudin Chapter 3 Exercise 14 (e)
The question posed is as follows:
For $\{s_n\}$ a sequence of complex numbers, $\sigma_n = \frac{s_0 + s_1 + ... + s_n}{n+1}$, $a_n = s_n - s_{n-1}$, $| n a_n | \leq M < \infty$, $ \forall n \in \...
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Let $a_n$ and $x_n,y_n\ge0$ be sequences such that $(x_na_n)$is Cesaro summable, $mx_n\le y_n\le Mx_n$ for some $m,M>0$ , $|x_na_n|\le1$ and $x_n\to0$
Let $(a_n),(x_n)$ and $(y_n)$ be sequences of real numbers with $x_n,y_n\ge0$, $mx_n\le y_n\le M x_n$ for some $m,M>0$, $|x_n a_n|\le1$, $x_n\to0$ and $\lim\limits_{N\to\infty}\frac{1}{N}\sum\...
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Convergence speed of Cesaro mean
Consider a sequence $(a_n)$ satisfying $\lim_{n\to\infty} a_n = a$. Let $b_n = \frac{1}{n} \sum_{i=1}^n a_i$. I have already known that $\lim_{n\to\infty} b_n = a$. I am wondering is there any ...
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Is the average of two convergent series equal to the Cesàro sum of the alternating series?
If we have $(a_n)$ and $(b_n)$ such that $\sum a_n$ converges and $\sum b_n$ converges, I know that we do not necessarily have that $\sum c_n$ (where $(c_n)_n=a_0,b_0,a_1,b_1,\dots$ converges.
But is ...
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Convergence of Power Series to Its Cesaro Sum sentence in Fourier proof
Well, I am learning about Fourier sum, and I encountered Cesaro sum in the proof of convergent uniformly of Fourier sum,
I know that Fejér sentence says that:
$\|f(x) - \sigma_n(f))\| < \epsilon .$ ...
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Let $c_n\ge0$ be a sequence. Sufficient conditions of $(c_n)$ such that $\lim 1/n\sum\limits_{k=1}^n c_k z^k=0$ for $|z|=1,z\ne 1$
Let $(c_n)$ be a sequence of non-negative reals which is bounded below and above i.e. $m\le c_n\le M$ for some $m,M>0$. But this is not enough to say about the limit of $\frac{1}{n}\sum\limits_{k=1}...
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What is the Cesaro sum of $(-1+1-1+1-1\ldots )$?
I have recently familiarized myself with the peculiar result of:
$1-1+1-1+1\ldots=\frac{1}{2}$
Following this enlightment I was now interested in finding out whether the following infinite series has ...
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