Lp approximation by quasi-convex and convex functions

Quasi-convex Functions and Quasi-monotone Operators

The notions of a quasi-monotone operator and of a cyclically quasi-monotone operator are introduced, and relations between such operators and quasi-convex functions are established.

Some Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points

some properties of certain subclasses of close-to-convex and quasi-convex functions with respect to 2k-symmetric conjugate points

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Convex Polynomial and Spline Approximation in Lp , 0 < p <

We prove that a convex function f 6 L p [ -1 , 1], 0 < p < cx~, can be approximated by convex polynomials with an error not exceeding Cw~ (f, 1/n)p where w~(f, .) is the Ditzian-Totik modulus of smoothness of order three of f . We are thus filling the gap between previously known estimates involving og~(f, I/n)p, and the impossibility of having such estimates involving ~o4. We also give similar...

Global and Fine Approximation of Convex Functions

Let U ⊆ R be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function f : U → R can be approximated by real analytic convex functions, uniformly on all of U . We also show that C-fine approximation of convex functions by smooth (or real analytic) convex functions on R is possible in general if and only if d = 1. Nevertheless, for d ≥ 2 we give a characterizat...