Shape-preserving approximation inL p

We prove a direct theorem for shape preservingL_p-approximation, 0p, in terms of the classical modulus of smoothnessw₂(f, t_p¹). This theorem may be regarded as an extension toL_p of the well-known pointwise estimates of the Timan type and their shape-preserving variants of R. DeVore, D. Leviatan, and X. M. Yu. It leads to a characterization of monotone and convex functions in Lipschitz classes (and more general Besov spaces) in terms of their approximation by algebraic polynomials.Communicated by Ron DeVore.