Asymptotics of the Lebesgue constants for bivariate approximation processes

TL;DR

This paper derives asymptotic formulas for Lebesgue constants associated with specific bivariate approximation processes, including Lagrange interpolation and Fourier partial sums, based on special node points and geometric configurations.

Contribution

It provides new asymptotic formulas for Lebesgue constants in bivariate approximation, focusing on Lissajous-Chebyshev nodes and anisotropic Fourier partial sums.

Findings

Asymptotic formulas for Lebesgue constants are established.

Results apply to Lagrange interpolation on Lissajous-Chebyshev nodes.

Findings enhance understanding of approximation behavior in bivariate Fourier analysis.

Abstract

In this paper asymptotic formulas are given for the Lebesgue constants generated by three special approximation processes related to the $\ell_1$-partial sums of Fourier series. In particular, we consider the Lagrange interpolation polynomials based on the Lissajous-Chebyshev node points, the partial sums of the Fourier series generated by the anisotropically dilated rhombus, and the corresponding discrete partial sums.