Let $f:(0,1) to mathbb R$ be a function whose fourier series can be written as $$f = sum_{k=1}^infty c_k phi_k,$$ where $phi_k$ are the eigenfunctions of the Laplacian with Neumann boundary conditions. What is the minimal assumption required on $f$ to obtain

$$

sum_{k=1}^infty |c_k| < 1 ?

$$

I am aware of the estimate $|c_k| lesssim k^{-n}$ if $f in C^n$, but I’m looking for something more precise as above.