A Fast, Simple, and Stable Chebyshev--Legendre Transform Using an Asymptotic Formula

A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree $N$ polynomial in $\mathcal{O}(N (\log N)^2/\log \log N)$ operations is derived. The fundamental idea of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the discrete cosine transform. Numerical results are provided to demonstrate the efficiency and numerical stability. Since the algorithm evaluates a Legendre expansion at an $N+1$ Chebyshev grid as an intermediate step, it also provides a fast transform between Legendre coefficients and values on a Chebyshev grid.