ABSTRACT This paper introduces a unified framework in Hilbert spaces for applying high‐order differential operators in bounded domains using Chebyshev, Legendre, and Fourier spectral methods. By exploiting the banded structure of differentiation matrices and embedding boundary conditions directly into the operator through a scaling law relating functions to their derivatives, the proposed approach achieves optimal matrix conditioning, thereby enhancing numerical stability for high‐order operators. Furthermore, it ensures consistent nodal and modal representations across Chebyshev, Legendre, and Fourier bases, consolidating similarity transformations. The method provides high accuracy for problems with inhomogeneous boundary conditions, eliminating the need for a priori polynomial factorization, and offers a generalized approach applicable to multi‐point boundary value problems. Finally, an error bound estimation is presented using backward consistency analysis. The methodology is validated through theoretical analysis and numerical experiments, demonstrating its robustness and accuracy for high‐order boundary value problems.
Guimarães et al. (Thu,) studied this question.