Questions tagged [tauberian-theorems]
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35 questions
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Lower bounds for $\|f*g\|_1$ with mean-zero Lipschitz functions on $[0,1]$
Let $f,g \in L^{1}([0,1])$ satisfy
$$
\|f\|_{1}=\|g\|_{1}=1, \qquad \int_{0}^{1} f(x)\,dx=\int_{0}^{1} g(x)\,dx=0,
$$
and assume
$$
f \in \mathrm{Lip}_{L_f}, \qquad g \in \mathrm{Lip}_{L_g}.
$$
...
0
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0
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58
views
Reference Request: Second Order Tauberian Theorem
I am interested in whether there exist Tauberian theorems for deducing the second order asymptotics of a particular function.
Throughout this post I am using the notation from Kwaśnicki's answer to ...
- 101
9
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1
answer
576
views
Is this "worst-case" example of the effect of poles of Dirichlet series on $\Re s =1$ known?
Let $A(s) = \sum_n a_n n^{-s}$. A standard example showing that you cannot deduce $\sum_{n\leq x} a_n = (1+o(1)) x$ just from a real Tauberian theorem is $a_n = \cos(\log n)$. We can easily modify ...
- 22k
5
votes
1
answer
365
views
Equivalent statement of the Wiener-Tauberian theorem?
I would like to know why we have the equivalence between the following statements of the Wiener-Tauberian theorem:
Version 1: Every proper closed ideal in $L^1(\mathbb R)$ is contained in a maximal ...
- 322
1
vote
0
answers
80
views
Reference request: $C_0$ version of Wiener's Tauberian theorem
Norbert Wiener has written 2 version of his Tauberian theorem for translations of functions: one for $L^1$ and one for $L^2$. I am wondering whether a similar statement exists for uniformly continuous ...
- 181
1
vote
0
answers
90
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A Tauberian limit
Let $\{a_n\}$ be a nonnegative real sequence and $\delta>0$. Set
$$A_N=\sum_{n=0}^Na_n,\quad B_N(\delta)=\sum_{n=1}^{N\delta} a_{N+n}\left( 1-\frac{n}{N\delta} \right),\quad c_N=1-e^{-1/N}.$$
...
- 501
1
vote
1
answer
353
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About Tauberian theorem : a condition of convergence on Cesaro sequence
During my research I came across this question.
Let $(u_n)$ a real sequence, with $U_n=\dfrac 1 n \sum\limits_{k=1}^nu_k$ converge to $l$, and $\exists N \in \mathbb N, N>10, \forall n \in \mathbb ...
- 6,007
5
votes
2
answers
367
views
Residue of Dirichlet series at $s = 1$
Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and suppose that the sequence has a well-defined "average", in the sense that
$$ \lim_{N \to \infty} \frac{1}{N}\sum_{i = 1}^N a_i = R$...
- 38.7k
0
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Validity of a Tauberian theorem for Dirichlet series
I encountered this statement about Dirichlet series but couldn't find a similar result in Korevaar's "Tauberian Theory". Is this statement valid?
Statement:
Let $f(s) = \sum_{n=1}^{\infty} \...
- 703
5
votes
0
answers
137
views
Possible extension of Ikehara's theorem for Dirichlet series with not necessarily positive coefficients?
I'm wondering about a possible extension of Ikehara's theorem for Dirichlet series with coefficients that are not necessarily positive. Consider the following:
Let $D(s) = \sum_{n \geq 1} \frac{a(n)}{...
- 703
2
votes
0
answers
283
views
A sharp version of a Tauberian theorem
The following Tauberian theorem is true (see Theorem I.11.1 of ''Tauberian theory: A century of developments''). Let $ a_n $ a sequence of real numbers.
If $f(x) = \sum_{n=1}^\infty a_n x^n $ ...
0
votes
1
answer
290
views
Explanation for Tauberian theorems for Laplace transform
I am struggling with the following theorem in Feller's book "Probability Theory and its Applications". The tauberian theorem is written as follow :
Let $F : [0,\infty) \to \mathbb{R}$ of ...
- 403
1
vote
1
answer
280
views
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 ...
- 2,384
3
votes
2
answers
578
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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)$.
- 22k
6
votes
1
answer
573
views
(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)=...
- 22k