# Questions tagged [cesaro-summable]

For questions about Cesàro summation and Cesàro summable sequences.

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### Cesaro summation of the inverse Fourier transform.

Let $f\in \mathcal{L}(\mathbb{R}^1).$ Prove that $$f(x)=\frac{1}{\sqrt{2\pi}}\lim_{T\to+\infty}\frac{1}{T}\int_{0}^{T}\int_{-t}^{t}e^{ixy}\hat fdy\,dt$$ for almost all x including points of continuity ...
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### Proving that if Sequence Converge then Series also Converge [closed]

Prove using Stolz–Cesàro theorem If $\lim\limits_{n→∞}a_n＝α$ then, $\lim\limits_{n→∞}\frac{S_n}{n}＝α$ I have tried using epsilon but I can not figure it out. Can someone please elaborate.
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### Equivalence on Cesàro convergence

Let $(x_n)$ be a bounded real sequence. Then $$\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n x_i=0 \Longleftrightarrow \lim_{n\to \infty}\frac{1}{2^n}\sum_{i=2^n+1}^{2^{n+1}} x_i=0.$$ Do you have a ...
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Let $(e_n)_{n\in\mathbb{\mathbb{Z}}} = (t\mapsto e^{int})_{n \in \mathbb{Z}}$ the Fourier orthonormal base of $L^2(\mathbb{T})$ with the scalar product $\langle f,g\rangle = \int_{-\pi}^\pi f(t)\bar{g}... • 4,628 0 votes 0 answers 44 views ### Generalized Cesàro summability of$(-1)^nn^p$A sequence$\{a_n\}_{n\geq 0}$is said to be$(C,\alpha)$-summable if$\lim_{n\to\infty} S^\alpha_n$exists, where $$S^n_\alpha = \sum_{k=0}^n \frac{ {n \choose k} }{ {n+\alpha \choose k} } a_k.$$ ... • 329 4 votes 3 answers 176 views ### Proof that$\sum_{n=0}^{\infty}(-1)^{n} = \frac{1}{2}$. Is there any error? So, I proved that: $$\int f(\ln x)\ dx = x \sum_{n=0}^{\infty}(-1)^{n} f^{(n)}(\ln x) \ \ \ +\ \ C$$ where$f^{(n)}$is the nth derivative of$f$. if we let$f(x) = e^{x}$then$f^{(n)}(x) = e^x$as ... • 3,517 1 vote 0 answers 132 views ### Cesàro Mean of Convergent Subsequence Converges? Let$X$be a compact and convex subset of$\mathbb R^n$and let${(x_{t})}_{t}$be a sequence in$X$such that$||x_{t+1}-x_{t}||_{\infty}\leq \frac{1}{t+1}$for all$t=0,1,...$. Let${(x_{t_n})}_{...
Loosely, I would like to know "how many" binary sequences are non-convergent in the sense of their average. Let $\{0,1\}^{\mathbb N}$ denote the set of all binary sequences. This set is homeomorphic ...