A linear barycentric rational interpolant on starlike domains

When an approximant is accurate on an interval, it is only natural to try to extend it to multidimensional domains. In the present article we make use of the fact that linear rational barycentric interpolants converge rapidly toward analytic and several-times differentiable functions to interpolate them on two-dimensional starlike domains parametrized in polar coordinates. In the radial direction, we engage interpolants at conformally shifted Chebyshev nodes, which converge exponentially for analytic functions. In the circular direction, we deploy linear rational trigonometric barycentric interpolants, which converge similarly rapidly for periodic functions but now for conformally shifted equispaced nodes. We introduce a variant of a tensor-product interpolant of the above two schemes and prove that it converges exponentially for two-dimensional analytic functions—up to a logarithmic factor—and with an order limited only by the order of differentiability for real functions (provided th...