A note on convex approximation in \(L_ p\) (Q1890593)

This note provides estimates for the approximation of a bounded convex function \(f\in L_ p[- 1, 1]\), \(1\leq p\leq \infty\), by convex polynomials. The estimates involve the second Ditzian-Totik modulus of smoothness of \(f\) in the sup-norm, hence one reason for the requirement of boundedness of \(f\). With \(\varphi:= \sqrt{1- x^ 2}\) and \(q\) satisfying \({1\over q}+ {1\over p}= 1\), it shows in a constructive way, that for each \(n\geq 1\) a convex polynomial \(P_ n\) exists, such that \[ \| f- P_ n\|_ p\leq Cn^{- 2/p} \omega^ \varphi_ 2(f, 1/n)^{1/q}_ \infty, \] where \(C= C_ 0 \delta(f)^{1/p}\), \(C_ 0\) an absolute constant and \[ \delta(f):= \sup_{x\in [- 1,1]} f(x)- \inf_{x\in [- 1,1]} f(x). \] Note that if \(f\in C[- 1,1]\) (that is if \(f\) is also continuous at the end points \(\pm 1\)), then for \(1< p< \infty\) we get convex approximation with accuracy of \(o(n^{- 2/p})\). For \(p= 1\) we have \(O(n^{- 2})\) which is best possible. Better estimates are valid for \(p= \infty\).