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Consider a Function $f\in L^2(\mathbb{T})$. Is there any lower bound for the decay of the Fourier coefficients

$$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt$$ known?

There are a lot of upper bounds known but i cant find anything about a lower bound.

I would appreciate if you can help me!

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    What do you exactly mean by lower bound?2012-06-05
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    BTW welcome to Math.SE!2012-06-05
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    I mean the following: $ |\hat f(n)|\ge g(n)$ for all $n\in \mathbb{N}$, where $g\in o(n!)$ for example.2012-06-05
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    more precisely i am concerned about the coefficients of a function $f^{-1}$, where f is a polynomial.2012-06-05
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    you can invert them more or less explicitly, the case of roots with $\vert z_0 \vert < $ or $ >0$ slightly different. you often see this done in time series classes. The resut is coefficients of $\frac 1 f $ exponential in root or reciprocal roots.2012-06-05
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    actually you're probably better off using countour integration to get the asymptotocs, tho the result is the same: that they are controlled by the closest zeros to the unit circle.2012-06-05
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    So, $f$ is the reciprocal of a (trigonometric or algebraic?) polynomial. This information certainly belongs in the post, because the question is trivial ("$0$ is the best lower bound you can have) without such information about $f$. As it stands, we still don't know enough to give any nontrivial bound: if $f=[1+\text{(some tiny polynomial terms)}]^{-1}$, then $\hat f(n)$ is tiny for $n\ne 0$.2012-06-05

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