The asymptotic Chebyshev coefficients for functions with logarithmic endpoint singularities: mappings and singular basis functions

Abstract

When a function is singular at the ends of its expansion interval, its Chebyshev coefficients a n converge very poorly. We analyze three numerical strategies for coping with such singularities of the form (1± x) k log(1± x), and in the process make some modest additions to the theory of Chebyshev expansions. The first two numerical methods are the convergence-improving changes of coordinate x=sin[( π/2) y] and x= tanh[Ly⧸(1-y 2) 1 2 ] . We derive the asymptotic Chebyshev coefficients in the limit n → ∞ for both mappings and for the original, untransformed Chebyshev series. For the original function, the asymptotic approximation for general k is augmented by the exact Chebyshev coefficients for integer k. Numerical tests show that the sine mapping is excellent for k⩾1, increasing the rate of convergence to b n = O(1⧸ n 4 k+1 ). Although the tanh transformation is guaranteed to be better for sufficiently large n, we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice: “sufficiently large n” is usually huge. Instead of mapping, one may use a third strategy: supplementing the Chebyshev polynomials with singular basis functions. Simple experiments show that this approach is also successful.