# Questions tagged [tauberian-theory]

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15 questions
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### 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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### 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 $$f(r)=\sum a_nr^n$$ ...
1answer
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### 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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### 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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### 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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### 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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### 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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### 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=\...