Norm in the space of absolutely convergent Fourier series

Question

In the book Ensemble parfaits et séries trigonométriques, Kahane and Salem assert the following :

$\big\|\exp\big(ing(x)\big)\big\|_{A(\mathbb{T})}$ is $O(|n|)$ where $g \in C^\infty(\mathbb{T})$ is real-valued and $n \in \mathbb{N}$.

Here the $A(\mathbb{T})$-norm is of course the norm of the Wiener algebra of functions with absolutely convergent Fourier series.

How come this norm is $O(|n|)$ ?

The $C^1$-norm of $\exp\big(ing(x)\big)$ is $O(|n|)$. Could this be of any use here ? — M.G, Nov 19 '16 at 18:10
A typo I meant $\mathcal{O}(k^2)$, and I'm not sure how you can improve it to $\mathcal{O}(k)$ — reuns, Nov 19 '16 at 18:36
To be clear, $g(x)$ is real ? In that case the https://en.wikipedia.org/wiki/Blaschke_product could help — reuns, Nov 19 '16 at 18:44
I couldn't find any indication that $g(x)$ is real but Kahane & Salem are merely explaining a proof by Kaufman. I just had a look to the article of Kaufman and he states that $g(x)$ is indeed real-valued. — M.G, Nov 19 '16 at 23:55
Take $g(x) = -i$, then $c_0(e^{i k g(x)}) = 2\pi e^k$ so yes obviously $Im(g(x)) \ge 0$ — reuns, Nov 20 '16 at 00:07
@M.G I see that you have created ([tag:wiener-algebra]) tag. It might be useful to create also tag-wiki or at least tag-excerpt. It might help other users to use the tag correctly. (This is probably not a problem here, since the tag name seems to be descriptive enough.) Another reason is that the tags used on only one question are automatically deleted after certain time unless they have tag-wiki. — Martin Sleziak, Dec 11 '16 at 06:16
I am a bit surprised to see that your suggested edit was rejected. While in theory I agree with the reason described there, in this case the name of the tag seem unambiguous enough to me. I wen ahead and edited it with the text from your suggestion. — Martin Sleziak, Dec 12 '16 at 01:08

reuns · Accepted Answer · 2016-11-19T18:40:45.310

Your claim is not true if $Im(g(x)) < 0$, in that case it grows exponentially. And if $Im(g(x))\ge 0$ I find $\mathcal{O}(k^2)$ :

$$\|f\|_{A(\mathbb{T})}= \sum_{n=-\infty}^\infty |c_n(f)|, \qquad \qquad c_n(f) = \int_0^{2\pi}f(x)e^{-i n x}dx$$

Also note that $$c_n(f')=i n c_n(f), \qquad\qquad c_n(f'')=-n^2 c_n(f)$$ With $G_k(x) = e^{ik g(x)}$ we have $G_k'(x) = ik g'(x)G_k(x)$ and $G_k''(x) = ik g''(x)G_k(x)-k^2g'(x)^2 G_k(x)$.

Assuming $Im(g(x))\ge 0$ we have $|G_k(x)|,|g(x)|,|g'(x)|,|g''(x)|$ bounded by a constant $\beta$, so that $|G_k''(x)|<k^2 \beta_2$,

and for $n \ne 0$ : $$|c_n(G_k)| = \frac{|c_n(G_k'')|}{ n^2} \le \frac{ k^2 \beta_2}{n^2}$$ with i.e.

$$\|G_k\|_{A(\mathbb{T})} = |c_1(G_K)|+\sum_{n\ne 0} |c_n(G_k)| \le 2\pi\beta+2\pi k^2 \beta_2\sum_{n\ne 0} \frac{1}{n^2}= \mathcal{O}(k^2)$$

Norm in the space of absolutely convergent Fourier series

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