Let $e_k$ be a complete orthonormal basis of $L^2([a,b])$ (for example, $[a,b]=[0,1]$, and the basis is the shifted Legendre polynomials, but I hope the answer to my question is not basis dependent or $[0,1]$ dependent).
Let $\{a_n\}$ be a sequence of real numbers. We know that $\{a_n\} \in l^2(\mathbb N) \iff \exists f \in L^2([a,b])$ such that $a_n = \langle f, e_n \rangle_{L^2}$, see Necessary and sufficient condition for a sequence of real numbers to be Fourier coefficients of some function
Now what subset of $l^2(\mathbb N)$ for the Fourier coefficients we would get if we added regularity on the function $f$ in the above equivalence ?
For example, $f$ is non-decreasing ? Or continuous ? Or continuous and non-decreasing at the same time ? Do we get better than square summability of the Fourier coefficients ? Or maybe the Fourier coefficients become a non-decreasing sequence (in the sense that they can be arranged in a non-decreasing way) ?