It is possible to approximate a function $f$ on $[0,2\pi]$ by a continuous function whose derivative is zero almost everywhere (as can be seen here : http://math.stackexchange.com/questions/67334/approximating-a-continuous-function-by-one-with-zero-derivative).

My question is as follows: knowing that $\|\cdot\|_\infty \leq \|\cdot\|_{A(\mathbb{T})}$, is it possible to strenghten this result ? i.e. is it possible to approximate $f$ in the $\|\cdot\|_{A(\mathbb{T})}$ norm by a continuous function whose derivative is zero almost everywhere ?

Here $\widehat{h}(n)$ denotes the $n$-th Fourier coefficient of $h$ and $$ \|h\|_{A(\mathbb{T})} :=~ \sum\limits_{n \in \mathbb{Z}}|\widehat{h}(n)|.$$

It is possible to approximate a function $f$ on $[0,2\pi]$ by a continuous function whose derivative is zero almost everywhere (as can be seen here : https://math.stackexchange.com/questions/67334/approximating-a-continuous-function-by-one-with-zero-derivative).

My question is as follows: knowing that $\|\cdot\|_\infty \leq \|\cdot\|_{A(\mathbb{T})}$, is it possible to strenghten this result ? i.e. is it possible to approximate $f$ in the $\|\cdot\|_{A(\mathbb{T})}$ norm by a continuous function whose derivative is zero almost everywhere ?

Here $\widehat{h}(n)$ denotes the $n$-th Fourier coefficient of $h$ and $$ \|h\|_{A(\mathbb{T})} :=~ \sum\limits_{n \in \mathbb{Z}}|\widehat{h}(n)|.$$