Fractional Inner Products and Orthogonal Polynomial Structures: A Riemann-Liouville Framework for Spectral Approximation

This paper develops an operator-oriented framework for spectral approximation in fractional calculus by introducing a fractional inner product defined through the Riemann-Liouville integral. Instead of modifying polynomial families, the proposed approach continuously deforms the underlying Hilbert space structure, with the fractional order α acting as a deformation parameter. A central theoretical result shows that this fractional inner product is mathematically equivalent to a classical weighted inner product with a deformed weight wα(x)=(b−x)α−1w(x). This equivalence establishes a rigorous connection between fractional calculus and classical orthogonal polynomial theory and clarifies the structural role of the fractional parameter. For a canonical one-dimensional setting, explicit recurrence relations are derived and the limiting behavior as α→1 is characterized, recovering the classical theory. The resulting orthogonal systems are naturally compatible with fractional operators and are used to construct spectral Galerkin methods for fractional differential equations. Well-posed variational formulations and optimal convergence rates are established. Numerical experiments illustrate the effectiveness of the framework, demonstrating spectral accuracy and improved performance in the approximation of fractional integrals and selected fractional differential equations when compared with standard polynomial bases. The proposed formulation provides a unifying operator-level perspective for spectral methods in fractional calculus.