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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 $\alpha > 0$

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.

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 $O(n^{-k})$.

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 $\alpha > 0$

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 $g:\mathbb{R} \to \mathbb{R}$.

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 $O(n^{-k})$.

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 $O(n^{-k})$.