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Questions tagged [wieners-tauberian-theorem]

Intended to label questions regarding Wiener's Tauberian Theorem and related topics.

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4
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1answer
68 views

Wiener's tauberian theorem for Hardy space

For $a>0$ let us define $$H^2(-a,a)=\{f \ \mbox{is analytic in the strip $|\Im(z)|<a$}: \sup_{y\in [-a,a]}\int_{\mathbb{R}}|f(x+iy)|^2\,dx<\infty\}.$$ For $f\in H^2(-a,a)$, define $\|f\|=\...
2
votes
1answer
39 views

How to generalize Newman's simplification of O-Tauberian theorem?

Can you prove the next theorem: Let $f$ be Dirichlet series with real, positive coefficients $(a_n>0)$. If $f$ is holomorphic on $\Re(z)\ge1$, but has one singularity at $z=1$, then $\lim_\limits{...
1
vote
0answers
41 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. ...
0
votes
1answer
110 views

The tauberian final value Theorem

I have found many formulations of the FVT (final value theorem), but none of this with the precise assumptions that one needs. As far I understood, one needs to assume that the $\lim_{t\to+\infty}f(t)...
2
votes
0answers
61 views

A Wiener-Ikehara variant with higher order poles

The problem: I am concerned of getting a generalisation of the Wiener-Ikehara theorem for Dirichlet series which are analytic in the plane $\{s\in\mathbb{C}:\sigma>1\}$ and extend analytically over ...
4
votes
1answer
138 views

Banach Algebra Question

Suppose $A$ is a commutative Banach algebra with identity and $I\subset A$ is an ideal. If there is a unique maximal ideal $M$ with $I\subset M$ does it follow that $I=M$? Or that $I$ is dense in $M$? ...