Best approximation by polynomials on the conic domains

TL;DR

This paper introduces a new modulus of smoothness and its equivalent K-function on conic domains in , characterizing weighted polynomial approximation and establishing direct and inverse theorems, especially addressing singularities at the apex.

Contribution

It defines a novel modulus of smoothness on conic domains and proves its equivalence to best polynomial approximation, including handling singularities at the apex.

Findings

Defined a new modulus of smoothness on conic domains.

Established direct and inverse approximation theorems using the modulus.

Characterized weighted polynomial approximation with a singularity at the apex.

Abstract

A new modulus of smoothness and its equivalent $K$-function are defined on the conic domains in $\mathbb{R}^d$, and used to characterize the weighted best approximation by polynomials. Both direct and weak inverse theorems of the characterization are established via the modulus of smoothness. For the conic surface $\mathbb{V}_0^{d+1} = \{(x,t): \|x\| = t\le 1\}$, the natural weight function is $t^{-1}(1-t)^\gamma$, which has a singularity at the apex, the rotational part of the modulus of smoothness is defined in terms of the difference operator in Euler angles with an increment $h/\sqrt{t}$, akin to the Ditzian-Totik modulus on the interval but with $\sqrt{t}$ in the denominator, which captures the singularity at the apex.