Abstract. The classical Weierstrass theorem states that any function continuous on a compact set $K\subset{\bf R}^d (d\ge1)$ can be uniformly approximated by algebraic polynomials. In this paper we study a possible extension of this celebrated result for approximation by {\it homogeneous} algebraic polynomials on {\it convex} surfaces ${K\subset\bf R}^d$ such that $K=-K$. Here we make a major progress in a previous conjecture proving that functions continuous on regular {\bf0}-symmetric convex surfaces can be approximated by a {\it pair} of homogeneous polynomials. Moreover, we settle completely the conjecture in $L_p$ metric when $1\le p< \infty $.

AMS Subject Classification (1991): 41A10, 41A63

Keyword(s): Jackson, uniform approximation, homogeneous polynomials, convex body

Received August 8, 2008, and in revised form November 17, 2008. (Registered under 6066/2009.)