On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev--Jacobi transform

We describe a fast, simple, and stable transform of Chebyshev expansion coefficients to Jacobi expansion coefficients and its inverse based on the numerical evaluation of Jacobi expansions at the Chebyshev--Lobatto points. This is achieved via a decomposition of Hahn's interior asymptotic formula into a small sum of diagonally scaled discrete sine and cosine transforms and the use of stable recurrence relations. It is known that the Clenshaw--Smith algorithm is not uniformly stable on the entire interval of orthogonality. Therefore, Reinsch's modification is extended for Jacobi polynomials and employed near the endpoints to improve numerical stability.