Endpoint-singularity-preserving spectral approximation theory for weakly singular integral equations

TL;DR

This paper develops a spectral approximation framework using fractional polynomials tailored for functions with endpoint singularities, enabling high-order solutions for weakly singular integral equations.

Contribution

It introduces a novel basis combining Jacobi polynomials with endpoint mappings, establishing structural properties and error estimates for singular problems.

Findings

Established orthogonality and derivative identities for fractional polynomials.

Proved projection and interpolation error estimates in weighted Sobolev norms.

Provided a rigorous foundation for spectral methods solving endpoint-singular problems.

Abstract

We introduce a fractional approximation framework for functions with limited regularity near the terminal point. The proposed basis is constructed by composing classical Jacobi polynomials with an endpoint algebraic mapping, thereby incorporating the terminal singular structure directly into the approximation space. The main structural properties of the fractional polynomials are established, including orthogonality relations, derivative identities, and a singular Sturm--Liouville eigenvalue formulation. We then introduce the associated weighted Sobolev spaces and prove projection and Gauss-type interpolation error estimates in weighted norms. Inverse inequalities and weighted Sobolev embedding estimates are also derived. The resulting theory provides a rigorous foundation for high-order spectral and collocation approximations of endpoint-singular and weakly regular problems, including terminal value problems, fractional differential equations, and weakly singular Volterra integral equations.