Questions tagged [tauberian-theorems]
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23
questions
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Relating $f(x)$ to its Laplace Transform for values other than $x=0$?
Suppose $X\in (0,1]$ is a random variable where $f(x)$ is its CDF and $g(t)$ is the Laplace Transform of $f(x)$. Tauberian theorems (Theorem 2.3 in Coqueret's "Approximation of probabilistic ...
- 2,270
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2
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396
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Bounds for PNT from Wiener-Ikehara?
What sort of bounds on the error term in the Prime Number Theorem can one obtain through a Wiener-Ikehara approach?
Same question, but for the Mertens function $M(x)=\sum_{n\leq x} \mu(n)$.
- 18.9k
6
votes
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answer
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(Explicit) Tauberian theorems: removing $(\log x/n)$
Say that $\{a_n\}_{n\geq 1}$, $|a_n|\leq 1$, are such that $$\left|\sum_{n\leq x} a_n \log \frac{x}{n}\right|\leq \epsilon x\quad\text{for all $x\geq x_0$.}$$ What sort of bound can we deduce on $S(x)=...
- 18.9k
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Hardy–Littlewood Tauberian theorem for Laplace transform
The Hardy–Littlewood Tauberian theorem for Laplace transform in Chapter XIII in "An Introduction to Probability Theory and Its Applications" by Feller reads as follows
Let $F : [0,\infty) \...
- 208
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Tauberian theorem with flatness condition
Suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is a series with $a_n\in \mathbb{R}$ and radius of convergence $1$ and such that $f$ restricted to $[0,1[$ admits a smooth extension to $[0,1]$ with $f^{(n)}(1)...
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Karamata's Abelian/Tauberian Theorem in the complex plane
The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels):
Fix $c, \rho>0$. If ...
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Tauberian lower bound for a series
Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of positive number such that $\sum_n a_n < +\infty$ (i.e. $a_n \in \ell^1$) but $\sum_n r^n a_n = +\infty$ for every $r > 1$.
Given $\sigma \in (0,1)$, ...
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Convergence speed of the tail of distribution using Tauberian remainder theorem
This question may be related to this one.
Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem.
Let $f$ be some one-sided probability ...
- 284
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Modern treatment of Delange's Tauberian Theorem
Tauberian theorems abound in the literature. One of the most general, powerful, and versatile is due to Delange, and appears as Theorem I of the paper:
H. Delange - Généralisation du théorème de ...
- 20.2k
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Tauberian theorem converse (Wiener-Ikehara)
Jacob Korevaar provides a nice converse to to Wiener-Ikehara tauberian theorem on p. 125 of his Tauberian Theory book:
For the non-decreasing, locally of bounded variation function $s$, if we have $\...
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Can the Selberg-Delange method be extended to analyzing $\sum_{n<x}\frac{a_n}{n}$?
The famous Selberg-Delange method takes sequences $a_n$ whose associated DGF $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ has a representation
$$F(s)=G(s;z)\zeta^z(s)$$
where $G(s;z)$ is "nice ...
- 2,591
0
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Reference needed for proof of a Tauberian theorem
I was reading the following paper by Hubert Delange: http://www.numdam.org/article/ASENS_1956_3_73_1_15_0.pdf 1. In page 26, he provides a proof of Theorem b, the bulk of which relies on a result in ...
- 709
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Dense stratification of a separable Hilbert space
Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps ...
- 4,971
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Weak version of Karamata's Tauberian theorem
I first posted this on mathematics. However, I got no answer there and it seems adapted here too. Also, it seems to be harder than I first thought.
Karamata's Tauberian theorem states the following. ...
- 1,835
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Relaxed/Truncated Version of Wiener's Tauberian Theorem
Background
Let $(U_t)_{t \in \mathbb{R}}$ be the (translation) $C_0$-group on $L^1(\mathbb{R})$ defined by
$$
U_t(f)(x) = f(x-t) \quad \text{for almost every } x \in \mathbb{R}
$$
(for $t \in \...
- 4,971
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votes
2
answers
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Reference request: Extensions of Wiener's Tauberian Theorem
Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
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Karamata's proof of Hardy-Littlewood Tauberian theorem
I understand Karamata's proof of the Hardy-Littlewood Tauberian theorem as in http://individual.utoronto.ca/jordanbell/notes/karamata.pdf, but what on earth is the motivation behind Lemma 4 - i.e, ...
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A question concerning Tauberian theory
Let $D(s) = \sum_{n=1}^\infty a_n n^{-s}$ be a Dirichlet series with $a_n ≥ 0$ and abscissa of convergence $\sigma_a = 1$. Further, we assume that $D(s)$ is holomorphic in each point $\Re(s) = 1$ ...
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One question about $L^1(G/K)$ and its closed subalgebra of $K$-invariant functions $L^1(G)^{\sharp}$
Can someone please clarify explicitly why: "The smallest closed subspace of $L^1(G/K)$ containing $L^1(G/K)^{\sharp}$ and invariant under the (left) $G$-action, is the full space $L^1(G/K)$".
Where $...
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Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $
I am trying to prove or disprove
$$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$
where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
- 3,413
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236
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A Generalized Wiener-Ikehara Theorem with multiple poles on the line
One version of the Wiener-Ikehara Theorem says that if
$$
f(s) = \sum \frac{a(n)}{n^s}
$$
is a Dirichlet series with nonnegative coefficients that converges absolutely for $\text{Re}(s) > 1$ and ...
- 1,313
11
votes
2
answers
694
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Tauberian theorem with better error term
This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...
- 1,352
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6
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The Wiener-Ikehara approach to the PNT
Was providing an alternative proof of the PNT one of the main impulses that led to the discovery of the Tauberian theorem of Wiener and Ikehara or the other way around?
In any case, do you know who ...
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