Decay conjecture for spectral coefficients of smooth mesh functions

Let the assumptions for the Fokker–Planck operator hold, and let an orthogonal decomposition based on such an operator be given. For smooth functions $f^i:\mathcal{M}^i\rightarrow\mathbb{R}$ with $f^i\in C^k$, $i=1,\ldots,m$, write $f^i|_K$ for their mesh representations and let $\alpha_j^i=\langle f^i|_K,\psi_j\rangle$, $j=1,\ldots,N_h$, be their spectral coefficients. Suppose $\gamma_{j+1}\geq\gamma_j$, with the $\gamma_j$ independent of $f^i$. Decay conjecture. The coefficients satisfy

$$|\alpha_j^i|\leq \frac{C}{(\gamma_j)^k}\left\|(f^i)^{(k)}\right\|_{L^2},$$

depending on the smoothness degree $k$ of the functions. Quantifying this decay and proving the estimate for the discrete Fokker–Planck setting remain open; the paper gives a proof strategy based on one-dimensional Sturm–Liouville estimates and spectral convergence results.