We prove a direct theorem for shape preserving Lp-approximation, 0<p<∞, in terms of the classical modulus of smoothness w2(f, tp1). This theorem may be regarded as an extension to Lp 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.
- AMS classification: 41A10, 41A25, 41A29
- Degree of monotone approximation
- Polynomial approximation in L