POLYNOMIAL-APPROXIMATION IN L(P)(S) FOR P-GREATER-THAN-0

For a simple polytope S in R(d) and p > 0 we show that the best polyno mial approximation E(n)(f)(p) = E(n)(f)L(p(S)) satisfies E(n)(f)(p) le ss than or equal to C omega(S)(r) (f, 1/n)p, where omega(S)(r)(f, 1/n) (p) is a measure of smoothness of f. This result is the best possible in the sense that a weak-type converse inequality is shown and a reali zation of omega(S)(r) (f, t)p via polynomial approximation is proved.