  When an approximant is accurate on the interval, it is only natural to try to
extend it to several-dimensional 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 on
two-dimensional starlike domains parametrized in polar coordinates. In radial
direction, we engage interpolants at conformally shifted Chebyshev nodes, which
converge exponentially toward analytic functions. In 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, if the boundary is as
smooth. Numerical examples confirm that the shifts permit to reach a much
higher accuracy with significantly less nodes, a property which is especially
important in several dimensions.
