Questions tagged [tauberian-theorems]

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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) \...
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3 votes
0 answers
98 views

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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182 views

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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3 votes
1 answer
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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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6 votes
1 answer
114 views

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 ...
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8 votes
0 answers
396 views

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 ...
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4 votes
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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 ...
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0 votes
1 answer
163 views

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 ...
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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 ...
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2 votes
2 answers
418 views

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. ...
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3 votes
1 answer
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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 \...
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  • 5,124
10 votes
2 answers
412 views

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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  • 650
2 votes
0 answers
346 views

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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7 votes
1 answer
335 views

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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1 vote
0 answers
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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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  • 630
14 votes
3 answers
765 views

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{\...
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  • 1,416
6 votes
0 answers
213 views

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 ...
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11 votes
2 answers
635 views

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, ...
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  • 1,332
10 votes
6 answers
2k views

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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