On $L_p$-error of bivariate polynomial interpolation on the square

TL;DR

This paper provides estimates for the $L_p$-error in bivariate polynomial interpolation on Lissajous-Chebyshev nodes, demonstrating similar behavior to tensor product Chebyshev grid interpolation for various functions.

Contribution

It extends error analysis to Lissajous-Chebyshev nodes for a broad class of functions, including non-smooth ones, showing comparable performance to classical Chebyshev grid interpolation.

Findings

$L_p$-errors behave similarly to tensor product Chebyshev grid interpolation.

Estimates apply to functions of bounded variation in Hardy-Krause sense.

Results include non-smooth functions, broadening applicability.

Abstract

We obtain estimates of the $L_p$-error of the bivariate polynomial interpolation on the Lissajous-Chebyshev node points for wide classes of functions including non-smooth functions of bounded variation in the sense of Hardy-Krause. The results show that $L_p$-errors of polynomial interpolation on the Lissajous-Chebyshev nodes have almost the same behavior as the polynomial interpolation in the case of the tensor product Chebyshev grid.