Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to obtain $$ \sum_{k=1}^\infty |c_k| < 1 \ ? $$
As pointed out in the comments below, this is related to characterizing the functions in the unit ball of the Wiener algebra.
I am aware of the estimate $|c_k| \lesssim k^{-m}$ if $f \in C^m(0,1)$, but I'm looking for something more precise as above.