#### Volume 55, pp. 726-743, 2022.

## A linear barycentric rational interpolant on starlike domains

Jean-Paul Berrut and Giacomo Elefante

### Abstract

When an approximant is accurate on an interval, it is only natural to try to extend
it to multi-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 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 that the boundary enjoys the same order of differentiability). Numerical examples confirm that the shifts permit one to reach a much higher accuracy with significantly fewer nodes, a property which is especially important in several dimensions.

Full Text (PDF) [507 KB], BibTeX

### Key words

barycentric rational interpolation, trigonometric interpolation, Lebesgue constant, conformal maps, starlike domains

### AMS subject classifications

41A10, 42A15, 41A20, 65D05

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