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Let $f(x) = \sum_{n=0}^\infty f_ne^{inx} + \bar{f_n}e^{-inx}$ and $g(x) = \sum_{n=0}^\infty g_ne^{inx} + \bar{g_n}e^{-inx}$ where $f:\mathbb{R} \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$. Both functions are uniformly continuous.

Suppose $f$ and $g$ are in $C^k$ with $|f_n|,|g_n| = O(n^{-k})$. Suppose $$f\circ g(x) = h(x) = \sum_{n=0}^\infty h_ne^{inx} + \bar{h_n}e^{-inx}$$

Is there a better asymptotic approximation to $|h_n|$ than $O(n^{-k})$. Can you give a bound if they are 1-Holder continuous? What if they are $\alpha$-Holder Continuous for $\alpha > 0$

Is there anything better you can say about the Fourier coefficient decay of $f(f(x))$ or $f^k(x)$ in general.

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  • $\begingroup$ What do you mean with $f \circ g$ for $f$ defined on $\Bbb{R}$ or $\Im(x)\ge 0$ and $g$ complex valued $\endgroup$
    – reuns
    Commented Aug 10, 2019 at 2:40
  • $\begingroup$ I edited the question. $\endgroup$
    – Halbort
    Commented Aug 10, 2019 at 2:56
  • $\begingroup$ $f$ is real valued iff its Fourier coefficients satisfy $F_{-n} = \overline{F_n}$. Also the decay of the Fourier coefficients depends on the local smoothnesses, and locally the composition of $C^k$ functions surjects on the $C^k$ functions. $\endgroup$
    – reuns
    Commented Aug 10, 2019 at 3:05
  • $\begingroup$ I have added the negative terms. I had forgotten to. Is there some way to give a better bound like $O(n^{-k + \frac{1}{\log(n)}})$ I want to show that the decay of $f^p(x)$ is slower. $\endgroup$
    – Halbort
    Commented Aug 10, 2019 at 3:19
  • $\begingroup$ The coefficients of $h \in C^k$ are $o(n^{-k})$ and $\sum_n |H_n|^2 n^{2k} < \infty$. To get stronger bounds you need things like $a$-Hölder continuity of $h^{(k)}$ $\endgroup$
    – reuns
    Commented Aug 10, 2019 at 3:24

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