SOME ESTIMATES FOR CONVEX POLYNOMIAL APPROXIMATION IN L p

We prove a direct theorem for convex polynomial L p-approximation, 0 < p < 1, in terms of the classical modulus of smoothness! 3 (f; t) p. This theorem may be regarded as an extension to L p of the well-known pointwise estimates of the Timan type and their variants of R. DeVore, Y. K. Hu and D. Leviatan for convex approximation in L p. It leads to a characterization of convex functions in Lipschitz classes (and more general Besov spaces) in terms of their approximation by algebraic polynomials