Some spectral problems of a diffusion operator under Paley-Wiener-based high-order approximations

In this study, we acquired spectral results for the diffusion operator under higher-order approximations. We reconstruct the well-known techniques and derive the essential results for the presented problem. The spectral results for the diffusion operator with high-order approximations were evaluated, focusing on solutions in the Paley-Wiener space. Additionally, we consider theorems that involve solutions belonging to the Paley-Wiener space and the applications of Shannon's sampling theorem. We also examine and evaluate the diffusion operator under more general separable boundary conditions.