# fa.functional analysis – Precise estimate on the decay of the coefficients in Fourier series

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.