Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous-Chebyshev nodes

TL;DR

This paper derives bounds for the Lebesgue constant in multivariate polynomial interpolation on Lissajous-Chebyshev nodes, linking it to Fourier series and polyhedral sets, revealing logarithmic growth similar to tensor product grids.

Contribution

It introduces new bounds for the Lebesgue constant in multivariate interpolation on polyhedral sets using Lissajous-Chebyshev nodes, connecting it to Fourier series analysis.

Findings

Bounds depend on logarithms of polyhedral side lengths

Lebesgue constant growth matches tensor product Chebyshev grid behavior

Provides insight into the condition number for multivariate interpolation

Abstract

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.