Resolution-optimal exponential and double-exponential transform methods for functions with endpoint singularities

We introduce a numerical method for the approximation of functions which are analytic on compact intervals, except at the endpoints. This method is based on variable transforms techniques using particular parametrized exponential and double-exponential mappings, in combination with Fourier-like approximation in a truncated domain. We show theoretically that this method is superior to variable transform techniques based on the standard exponential and double-exponential mappings. In particular, it can resolve oscillatory behaviour using near-optimal degrees of freedom, whereas the standard mappings require numbers of degrees of freedom that grow superlinearly with the frequency of oscillation. We highlight this result with several numerical experiments. Therein it is observed that near-machine accuracy is achieved using a number of degrees of freedom that is between four and ten times smaller than those of existing techniques.