Some suggestions concerning the conjecture in: 'Tractable semi-algebraic approximation using Christoffel-Darboux kernel'

TL;DR

This paper improves the approximation rate of semi-algebraic and definable functions using Christoffel-Darboux kernels from the conjectured rate to a faster rational rate in the uniform norm, for more regular functions.

Contribution

It demonstrates that for semi-algebraic and definable functions, the approximation rate can be enhanced to a rational rate in the supremum norm, surpassing previous conjectures.

Findings

Approximation rate improved to rational rate in L-infinity norm.

Results apply to semi-algebraic and definable functions.

Enhancement over previous conjectured rate.

Abstract

In 'Tractable semi-algebraic approximation using Christoffel-Darboux kernel' Marx, Pauwels, Weisser, Henrion and Lasserre conjectured, that the approximation rate $\mathcal O(\frac 1 {\sqrt(d)})$ of a Lipschitz functions by a semi-algebraic function induced by a Christoffel- Darboux kernel of degree $d$ in the $L^1$ norm can be improved for more regular functions. Here we will show, that for semi-algebraic and definable functions the results can be strengthened to a rational approximation rate in the $L^\infty$ norm.