On the orthogonality of a system of solenoidal functions in a three-dimensional cube

Previously, we constructed a system of orthonormal functions (SOF) as a solution to a spectral problem for a fourth-order operator in a three-dimensional cube. Using a three-dimensional curl operator applied to SOF, we derived a system of solenoidal functions (SSF), which are crucial in the study of incompressible fluid dynamics and the theory of Navier-Stokes equations. However, the SSF obtained in this way did not possess the orthogonality property, which is often desirable in theoretical analysis and numerical applications. The main result of this work is the construction of a new system of solenoidal functions, based on the original SOF, which is shown to be almost orthogonal. This property makes the system suitable for use in spectral methods and other analytical approaches where near-orthogonality ensures better convergence and stability. The methodology developed in this study can be generalized to other types of boundary value problems involving higher-order differential operators and may contribute to the development of more efficient computational schemes in fluid mechanics.