Classical orthogonal polynomials as a pedagogical tool in functional analysis and spectral theory

This article analyzes Legendre, Laguerre, and Hermite polynomials as eigenfunctions of self-adjoint differential operators defined on Hilbert spaces of the type , with the goal of strengthening their conceptual understanding in advanced educational settings. By reformulating their differential equations as Sturm–Liouville problems, we identify the structural components that allow for a spectral interpretation: weight functions, domains, eigenvalues, and orthogonality. This unified framework presents these polynomials not only as formal solutions but also as pedagogical tools to explain key concepts in functional analysis, such as orthonormality, discrete spectra, and complete bases. The approach combines mathematical rigor with visual and comparative resources to support their integration into the teaching of upper-level mathematics and physics courses. It is concluded that the spectral interpretation of these polynomial systems can significantly enhance students' conceptual comprehension by connecting topics from linear algebra, differential equations, and functional spaces. The study proposes their inclusion as didactic resources in university-level mathematics education, promoting meaningful learning around abstract structures.