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Let $$f=\sum f_{s}\exp(2\pi isx)\in C^{(p-1)}[0,1]$$ and $$f^{(p)}\ in\ L_2[0,1]\ \ ( \sum\left|f_{s}\right|^{2}j^{2p}<\infty )$$ Does it imply that $f_s=O(s^{-(p+\psi)})$ for some $\psi>0.5$?

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up vote 4 down vote accepted

You seem to be under the impression that any complex valued sequence $(a_s)$ such that $\displaystyle\sum_s|a_s|^2$ converges must be such that $|a_s|=O(s^{-\psi})$ with $\psi>\frac12$ (and if you are not, please forgive me). But this is far from the truth for at least two reasons.

First, it is enough to ask that $|a_s|=O(s^{-1/2}(\log s)^{-\phi})$ for a large enough $\phi$ (and what large enough means here, I will let you discover), a condition which does not imply the existence of $\psi>\frac12$ such that $|a_s|=O(s^{-\psi})$.

Second, $(a_s)$ could be a lacunary sequence. Assume for instance that $a_s=0$ for every $s$ except the powers of $2$, in which case $a_s=1/\log s$. Then (I will let you check that) $\displaystyle\sum_s|a_s|^2$ converges although the condition $|a_s|=O(s^{-\psi})$ is false for every positive $\psi$.

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