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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 : 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 $$ \|h\|_{A(\mathbb{T})} :=~ \sum\limits_{n \in \mathbb{Z}}|\widehat{h}(n)|.$$

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