Best approximation by polynomials on the conic domains

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.