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Questions tagged [tauberian-theory]

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26 views

Remainder in the Wiener-Ikehara theorem

I am looking for a reference for a variant of the Wiener-Ikehara theorem (for Dirichlet series) giving result of the form $$ \sum_{n\leq X} a(n) = cX^a(\log X)^m + O\big(X(\log X)^{m-1}\big),$$ i.e. ...
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Typo in first Tauberian theorem in Wiener's “Tauberian Theorems”?

In Equation (0.0.7)-(0.09) of Norbert Wiener's Tauberian Theorems, it is stated that [...] if $$ \lim_{x\to 1-0}\frac{1}{1-x}\sum_{0}^{\infty}a_nx^n=A, $$ and $$ a_n= o(1/n), $$ then $$ \sum_{0}...
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1answer
76 views

Hardy's power series. Cesàro convergence

Doing my analysis homework i have come across the following power series known as Hardy's power series $$\sum\limits_{k=0}^{\infty}a_kx^k=\sum\limits_{k=0}^{\infty}(-1)^kx^{2^k}\mbox{ for x}\in[0,1],$$...
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1answer
88 views

A Tauberian theorem for a quotient of power series, the limit on the boundary

Take sequences $a_n, b_n \in \mathbb R_{>0}$ (or $\mathbb C$) such that the limit $$L = \lim_{N \to \infty}\frac{\sum_{n \leq N} a_n }{\sum_{n \leq N} b_n}$$ exists and such that the power series ...
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84 views

Laplace transform and tail probability

Let $X \ge 0$ be a non-negative random variable. I would like to know if the following statements are equivalent: $$ \lim_{\lambda \to 0^+} \frac{\mathbb{E} \left[X e^{-\lambda X}\right]}{\log (1/\...
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100 views

Fourier series: $\hat f(n)=O(1/n)$ and $f$ continuous implies uniform convergence?

Littlewood's Tauberian theorem: Let $a_n=O(1/n)$. (In particular, given any $0<r<1$, the power series $\sum a_nr^n$ converges.) If the function defined by the power series $$f(r)=\sum a_nr^n$$ ...
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1answer
42 views

Final Value Theorem Variant

Let $f:\mathbb{R}\to\mathbb{R}$ be given (possibly with some conditions to be added later?). Prove the following statement: $$ \lim_{x\to\infty}\frac{1}{x}\int_0^xdy\int_0^y\,dz\,f(z)=\lim_{s\to0}\...
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2answers
108 views

A couple of difficulties in Tauber theory lecture notes of prof. Yum-Tong Siu

Reading trough the lecture notes on Tauber theory of prof. Yum-Tong Siu I am a bit off right at the beginning. If someone could clarify the follow two steps in his proofs of Tauber's original 1897 ...
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1answer
487 views

Verifying $|F(r)| \geq \frac{1}{1-r}\log(\frac{1}{1-r}) $ and $|F(re^{i \theta})| \geq c_{q/r}\frac{1}{1-r}\log({\log(\frac{1}{1-r})})$

I'm attempting to take a Tauberian route in verifying the proposition in $(1)$ below, which is from Complex Analysis, by Elias M Stein and Rami M. Shakarchi. Let $F(z)$ be the following series: $$F(...
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178 views

Showing Abel Means of a Fourier Series Converge Uniformaly to $f$?

In the text "Fourier Analysis and Related Topics", i'm having trouble proving the following Theorem in $(3.5.5)$ utilizing Fourier Methods/Summability Methods. Also i'm not sure how to approach $(ii)$ ...
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2answers
242 views

Proving $\sin(x) + \sin(2x) + \sin(3x)+ \cdots +\sin(nx) = \frac{cos(1/2) - cos(n-1/2)}{2 \sin(1/2)2}$ via Ceasro Summation?

I'm having trouble proving the following conjecture in $(1.)$, via Fourier methods, my intial attack can be seen in $(2)$ $(1.)$ $$\sin(x) + \sin(2x) + \sin(3x)+ \cdots +\sin(nx) = \frac{cos(1/2) - ...
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135 views

Showing the Series $\sum_{n=1}^{\infty}c_{n}$ convergences and that it's abel summable to s?

In stein's Fourier Analysis text i'm attempting to verifying my proof that the $\sum_{n=1}^{\infty}c_{n}$ converges to a finite limit $s$ then the series is Abel summable to $s$. My initial attempt ...
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0answers
19 views

Uniform tauberian theorem

Let $\epsilon>0$. I wonder if the following is true: There exists $\lambda_0\in(0,1)$, such that for all $\lambda\leq\lambda_0$ and all $(a_n)\in [0,1]^\mathbb{N}$ there exists $N\in \mathbb{N}$ ...
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174 views

Differentiating an asymptotic power series

I am interested in theorems that allow me to differentiate a divergent asymptotic power series for $x \to \infty$. I have a function $f:\mathbb{R}\to\mathbb{R}$ that is differentiable for large enough ...
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81 views

Example and counter-examples for the tauberian theorem of Hardy-Littlewood

We know the following theorem: Theorem (Hardy-Littlewood). Let $(b_n)$ be a real sequence such that (i) $(nb_n)$ is bounded, (ii) $\displaystyle\lim_{x\to 1^-} \sum_{n=0}^\infty b_n x^n=\...