Let $\mathbb Z$, the set of integers, be a group with respect to addition and its dual group is the $\mathbb T = \{z\in \mathbb C : |z|=1\}$ , one dimensional torus. Put, $\ell^{1}(\mathbb Z)= \{g:\mathbb Z \to \mathbb C : \sum_{m\in \mathbb Z} |g(m)|<\infty \}$, and for $g\in \ell^{1}(\mathbb Z)$, we define its Fourier transform on $\mathbb T$ as follows:
$$\hat{g}(e^{i\theta})=\sum_{m\in \mathbb Z} g(m) e^{-im\theta} \ \ (e^{i \theta }\in \mathbb T).$$ We consider the space(sub space as we have restrict to real functions) of Fourier transforms, $$A_{\mathbb R}(\mathbb T)=\{f:\mathbb T \to \mathbb R : \exists \ g\in \ell^{1}(\mathbb Z) \text{such that} \ \hat{g}= f \}.$$ $A_{\mathbb R}(\mathbb T)$ is normed by the $\ell^{1}-$ normed on $\mathbb Z$: $$||f||:=||g||_{\ell^{1}(\mathbb Z)}.$$ We also note that $A(\mathbb T)$ is a Banach algebra under pointwise addition and multiplication.
Fix $r\in (0, \infty)$, and let $f\in A_{\mathbb R}(\mathbb T)$ such that $||f|| \leq r$ then $e^{if} \in A_{\mathbb R }(\mathbb T)$ and $||e^{if}||\leq \sum_{m=0}^{\infty}\frac{||if||^{m}}{m!} \leq e^{||f||}\leq e^{r}$.
Put, $S_{r}: = \{ f\in A_{\mathbb R}(\mathbb T) \ \text {with} \ ||f||\leq r \}.$
My question : Does there exists $f\in A_{\mathbb R}(\mathbb T)$ with $||f||=r$ such that $$||e^{if}||=e^{r}$$; if yes, what is $f$ ? How to prove, $\sup \{||e^{if}||: f\in S_{r}\}= e^{r}$ ? Any suggestion or some specific references , concerning this, will be fine.
(Note: If we allowed complex valued function in $A_{\mathbb R}(\mathbb T)$ then there is $f\in A(\mathbb T)$ such that $||e^{if}||= e^{||f||}$; for instance, take $f(t)= -i, \ \ (t\in \mathbb T )$)
My attempt cum guess-work: Given $\epsilon > 0 $, we choose a positive integer $n> 2r$, so large that $$e^{r}\{(1+\frac{r^{2}}{n^{2}})^{n}-1\}< \frac{\epsilon}{2},$$ and $$(1+\frac{r}{n})^{n}> e^{r}-\frac{\epsilon}{2}.$$
We choose a points $x_{1},..., x_{n}$ in $\mathbb Z$ (not 0), such that $$x_{k+1}\not = x_{1}+x_{2}+...+x_{k} \ \ (k=1,2,...,n-1).$$
For $x\in \mathbb Z$, we define $\delta_{x}:\mathbb Z \to \{0,1\}$ such that
$$ \delta_{x}(m)=\begin{cases} 1 & \text{if} \ m=x,\\ 0, & \text {if} \ x\not = m. \end {cases}$$
and also we define a function $\sigma_{k}:\mathbb Z \to \mathbb R$, as follows
$$\sigma_{k}=\frac{1}{2}(\delta_{x_{k}}+\delta_{-x_{k}}) \ \ (k=1,2,..,n).$$
and using this we define, $g:\mathbb Z \to \mathbb R$ as follows $$g(m)= \frac{r}{n}(\sigma_{1}+\sigma_{2}+...+\sigma_{n})(m) \ \ (m\in \mathbb Z).$$
We define, $f:\mathbb T \to \mathbb R$ as
$$f(e^{i\theta}):= \hat{g}(e^{i\theta})\ \ (e^{i\theta} \in \mathbb T).$$
We notice that, $||f||= ||g||_{\ell^{1}(\mathbb Z)}= r$.
I must prove: $||e^{if}||> e^{r}-\epsilon $; Form here I don’t know how to proceed!!!
(This idea of constructing the above function I took from the paper "Functions which operates Fourier transform, (1959)" by H. Helson, J.-P. Kahane, Y. Katznelson, W. Rudin ; Lemma 2.1; they have proved this for space of Fourier-Steiltjes transforms on locally compact group; but I guess, proof concrete case, $A_{\mathbb R}(\mathbb T)$ must be there some where, in the literature, I really don't know ,and may be some easy example as well possible; )
Thanks;-)
