New exponential variable transform methods for functions with endpoint singularities (Q2927838)

Using the Schwarz-Christoffel formula, two parameterized exponential changes of variables are constructed for mapping analytic functions from their original domain \((0,1]\) to \((-\infty,0]\), or from \((0,1)\) to \((-\infty, \infty)\). The transplanted function on either the infinite or semi-infinite interval is then approximated using domain truncation, followed by Chebyshev interpolation. A thorough analysis of the resulting approximation methods obtains exponential convergence rates for their \(\infty\)-norm error, as well as asymptotic bounds for their resolution power (expressed in terms of the number of degrees of freedom required to represent an oscillatory function to a fixed precision). The results demonstrate that, compared to existing variable transform methods, the new methods not only achieve similar convergence, but can also lead, subject to appropriate parameter choice, to superior resolution properties. Numerical experiments that verify the results are provided and several directions for future work are proposed.