Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous-Chebyshev nodes
For researchers in approximation theory, this provides theoretical bounds for the conditioning of interpolation on Lissajous-Chebyshev nodes, but the result is incremental as it mirrors known behavior from tensor product grids.
The paper derives upper and lower bounds for the Lebesgue constant of multivariate polynomial interpolation on Lissajous-Chebyshev nodes, showing that the bounds grow as a product of logarithms of side lengths, matching the behavior of tensor product Chebyshev grids.
To analyze the absolute condition number of multivariate polynomial interpolation on Lissajous-Chebyshev node points, we derive upper and lower bounds for the respective Lebesgue constant. The proof is based on a relation between the Lebesgue constant for the polynomial interpolation problem and the Lebesgue constant linked to the polyhedral partial sums of Fourier series. The magnitude of the obtained bounds is determined by a product of logarithms of the side lengths of the considered polyhedral sets and shows the same behavior as the magnitude of the Lebesgue constant for polynomial interpolation on the tensor product Chebyshev grid.