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Does there exists a piecewise smooth, periodic, and continuous function $f$ such that

$$\vert \hat{f}(n)\vert > \frac{1}{n^{1/3}}$$

for all $n > 10$?

Honestly, i don't know where to start and i've been stuck for at least a day now. Any hints on how to proceed? Thanks!

Eugene
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  • What is your definition of $\hat{f}(n)$? – Steven Clark Apr 26 '23 at 16:40
  • The complex fourier coefficient of f – Eugene Apr 26 '23 at 23:24
  • the Fourier series of any bounded variation function are $O(1/n)$ so you definitely cannot get such an example; for a continuous function you can get coefficients that are in $L^2$ and $(1/n)^{1/3}$ is not in $L^2$ so you cannot get such even if you omit the piecewise smooth and keep just continuous; on the other hand you can get $1/(\sqrt n \log n)$ for example – Conrad Apr 27 '23 at 01:00

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